%!PS-Adobe-2.0 %%Creator: dvips 5.528 Copyright 1986, 1994 Radical Eye Software %%Title: final.dvi %%CreationDate: Mon Jun 14 15:16:54 1999 %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX12 CMSS10 CMR10 CMTT10 CMR17 CMR12 CMR8 CMBX10 %%+ CMTI10 CMR6 CMR9 CMTI9 CMTT9 CMCSC10 CMSY10 CMSL10 CMMI10 CMSY8 %%+ CMMI8 CMMI6 CMBX9 CMBXTI10 CMEX10 CMSY6 CMMI9 %%EndComments %DVIPSCommandLine: dvips -o final.ps final.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1999.06.14:1516 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D }B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 9B9C1569015F2C1D2BF560F4C0D52257BACDD6500ABDA5ED9835F6A016CFC8F00B6C052ED76A 87856B50F4D80DFAEB508C97F8281F3F88B17E4D3B90C0F65EC379791AACDC162A66CBBC5BE2 F53AAD8DE72DD113B55A022FBFEE658CB95F5BB32BA0357B5E050FDDF264A07470BEF1C52119 B6FBD5C77EBED964AC5A2BBEC9D8B3E48AE5BB003A63D545774B922B9D5FF6B0066ECE43645A 131879B032137D6D823385FE55F3402D557FD3B4486BE79011D1F5BFAE5C1F476EE6F05EB1D2 CAEB269958B194521197B312FCCED4867F3C8FBD030BD715D8FFDA1DCD454B174E7A1A97B59F E770E67702519D9D9B23D61AC08424D555242A8CA08C49AEF300945D99B999A79CE74804AE6B FDE623F4463371442F6523A5F6CE19C839A708C025132E22C696C8CCADE45680E5197189D0F9 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9470586ADD9DE03892B9E0989A2B1AAB57FAD07E5D9F7A5CB510AC82459949A286E4FBCA5199 54F1EC6118A03538FBCD4C0CF85AAFEAAB0390AE0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSS10 %!PS-AdobeFont-1.1: CMSS10 1.0 %%CreationDate: 1991 Aug 20 17:33:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-61 -250 999 759}readonly def /UniqueID 5000803 def currentdict end currentfile eexec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275AA6F8D39283433935E56A740523E72FA631C1A9DA0C1E9400E4D4B200ABEF380000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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10:42:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /visiblespace put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /visiblespace put dup 33 /exclam put dup 34 /quotedbl put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /backslash put dup 93 /bracketright put dup 94 /asciicircum put dup 95 /underscore put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /braceleft put dup 124 /bar put dup 125 /braceright put dup 126 /asciitilde put dup 127 /dieresis put dup 128 /visiblespace put dup 160 /space put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueID 5000832 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /sterling put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FFD20326947B1065649B533EB5E9E3A88A87FE9879 18A687950B26F647D6E08BF14D983ED1D4A8CA1E2DA8D985F9442EB916F5B6ED0697AC7C33E1 E36A3BF460D34CE45F1631871097CB04F18E38894CF4AC1538EB19481311D24FE3BE7BEAA4A3 730E8B4831FE59D6D9CE2E46116B629C7BA2F9CE3ECBA2F43BC162A5A077CA1B2882A42AFDCE C3F4B75B5D63E0BC8E5DC95257766D8EA467AD9CBAF47BE60F797580CCED6884B3A68F70C91F 4FDF80FD00ED9139E7F480DC3A76AF72AD9B434187730BDFAEFE4CBFE5C7EDCAAF249204FD70 3011932E5A3C27BE468B7DAB69DAA18DBBB6335AB8DDFDC607961C7C02582763FA069D43563B A17704029945CF42FCB19CF78C51DF0EC4D851086D43C1AB38E865EF36865FFF3A08B01CBCB0 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 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5D0AB08E0AF286D5F1352F0E4D5B58774F0324B5479621488FAF503415333C9B232119542A24 54A72918BCBAE3628177414CE76975F786C68AB4FBE47B150539D9BD711C510A7BB87523437F CB820000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /sterling put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 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cleartomark %%EndFont %%BeginFont: CMTT9 %!PS-AdobeFont-1.1: CMTT9 1.0 %%CreationDate: 1991 Aug 20 16:46:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /visiblespace put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /visiblespace put dup 33 /exclam put dup 34 /quotedbl put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /backslash put dup 93 /bracketright put dup 94 /asciicircum put dup 95 /underscore put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 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Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueID 5000798 def currentdict end currentfile eexec 9B9C1569015F2C1D2BF560F4C0D52257BACEE583A5C939393E012707B47E0C1FA47D284A1EDC 9D01A497D772BCA8C543388E6DC0D1E2C4944740470E0914F65EFB0737B0851B2BA713A9A00B 36D07DA6BCB52920B9B59EFE587734027A3C5E6566AAD332FE6FBCCED1417802822A3B81D618 7875263D6BBDA04BBCF6E4870FEEAD60F104BB3C5A766610DD11AEA64A6B107B0B04439FA288 8B8CC39232BB83F7695ABA81F0260CD5248B9E649CD800BCB325468C79FFACB9CD3036F2ED7B B65DC6F12E33DD6D45B61272314F98990D08C0F8274E999F4B46AF4B5C5DA923683C960FACE5 0087915D2044D7211A18E0C22967EF89552662BA990E2DFB9267E09E354F41E74D44BD80A2D9 05D2EDD60D8829E820C6AB0FDF322964C1ECB6569818A465D5266D3366D748B412446E602F0C 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 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ECF830BAEE80AF84B6202A6157C59DEE52E07EBD580002579AB13A7A9F48B8170278053CA1ED 4A345C0E04582705D0BC924DE91C306449CB8541A355D252085CEE5824BB68408AE62CE2CE6D 8BA05C7394400D64EE59FB23A5B77B4DAC2F30A5324886F4C9CF3C2F6A0BA347DACD4388DAE9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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271DB64DA625559B31C7D422EEF1ED5A35CBF1F37958EA0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /sterling put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put 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0B2DFE4CC7A04F22A149000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /parenleftbig put dup 162 /parenrightbig put dup 163 /bracketleftbig put dup 164 /bracketrightbig put dup 165 /floorleftbig put dup 166 /floorrightbig put dup 167 /ceilingleftbig put dup 168 /ceilingrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 173 /angbracketleftbig put dup 174 /angbracketrightbig put dup 175 /vextendsingle put dup 176 /vextenddouble put dup 177 /slashbig put dup 178 /backslashbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 185 /floorleftbigg put dup 186 /floorrightbigg put dup 187 /ceilingleftbigg put dup 188 /ceilingrightbigg put dup 189 /braceleftbigg put dup 190 /bracerightbigg put dup 191 /angbracketleftbigg put dup 192 /angbracketrightbigg put dup 193 /slashbigg put dup 194 /backslashbigg put dup 195 /parenleftBigg put dup 196 /arrowdblbt put dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 4 /floorleftbig put dup 5 /floorrightbig put dup 6 /ceilingleftbig put dup 7 /ceilingrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 14 /slashbig put dup 15 /backslashbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 22 /floorleftbigg put dup 23 /floorrightbigg put dup 24 /ceilingleftbigg put dup 25 /ceilingrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 30 /slashbigg put dup 31 /backslashbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 36 /floorleftBigg put dup 37 /floorrightBigg put dup 38 /ceilingleftBigg put dup 39 /ceilingrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 44 /slashBigg put dup 45 /backslashBigg put dup 46 /slashBig put dup 47 /backslashBig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 63 /arrowvertex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 70 /unionsqtext put dup 71 /unionsqdisplay put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 74 /circledottext put dup 75 /circledotdisplay put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 85 /unionmultitext put dup 86 /logicalandtext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 93 /unionmultidisplay put dup 94 /logicalanddisplay put dup 95 /logicalordisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 106 /floorleftBig put dup 107 /floorrightBig put dup 108 /ceilingleftBig put dup 109 /ceilingrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put dup 119 /arrowvertexdbl put dup 120 /arrowtp put dup 121 /arrowbt put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put dup 126 /arrowdbltp put dup 127 /arrowdblbt put dup 128 /parenleftBigg put dup 160 /space put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 80347982AB3942D930E069A70D0D48311D7190FA2D133A583138F76695558E7AE9348D37CAC6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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y(Switzerland,)13 b Fn(stege@inf)o(.e)o(thz)o(.c)o (h)k(.)966 2658 y Ft(1)p eop %%Page: 2 2 2 1 bop 228 213 a Fx(the)18 b(practical)f(folks)h(p)q(ortra)o(y)o(ed)f (in)i(the)f(am)o(using)g(carto)q(ons)f(of)h(Garey)g(and)g(John-)228 267 y(son)j([)p Fs(GJ79)p Fx(],)h(Chapter)g(1,)g(ha)o(v)o(e)f(b)q(een)i (somewhat)d(disapp)q(oin)o(ted)i(with)f(theoret-)228 321 y(ical)g(computer)g(science,)j(and)e(there)g(ha)o(v)o(e)f(b)q(een)i (n)o(umerous)f(inciden)o(ts)f(of)h(\\rest-)228 374 y(less)16 b(drums")g(in)g(the)g(jungles)g(of)g(computer)g(applications.)22 b(Suc)o(h)17 b(grum)o(blings)e(ma)o(y)228 428 y(ha)o(v)o(e)i(something) g(to)g(do)g(with)g(the)h(recen)o(t)f(spate)h(of)f(soul-searc)o(hing)f (among)h(theo-)228 482 y(rists)d([)p Fs(HL92,)j(PGWRS96,)g(AF)o(GPR96)p Fx(].)i(Some)c(examples:)228 590 y Fs(Example)21 b(1:)33 b(Y)l(et)22 b(Another)f(Call)h(for)g(Reform.)31 b Fx(One)20 b(of)f(the)h(plenary)f(ad-)228 644 y(dresses)d(at)g(the)g(AAAI)h (meeting)f(in)g(1996)f(w)o(as)g(concerned)i(with)f(the)g(broad)g(theme) 228 698 y(of)i(ho)o(w)f(computer)h(science)h(practice)f(and)g(theory)g (in)o(teract)f([)p Fs(DKL96)p Fx(].)29 b(The)18 b(dis-)228 752 y(cussion)d(in)g([)p Fs(DKL96)p Fx(])g(can)g(b)q(e)h(summarized)f (as:)228 806 y(\(1\))e(P)o(oin)o(ting)f(to)h(a)g(particular)g(problem,) g Fm(STRIPS)j(Planning)p Fx(,)d(as)g(cen)o(tral)g(to)g(the)228 860 y(en)o(tire)h(\014eld)i(of)f(arti\014cial)e(in)o(telligence.)228 914 y(\(2\))f(Prop)q(osing)h(that)f(practitioners)g(and)h(theorists)f (collab)q(orate)g(in)h(an)g(in)o(tense)g(ana-)228 968 y(lysis)j(of)h(this)g(one)g(problem,)g(to)g(understand)h(what)e(mak)o (es)h(it)g(hard,)g(and)h(to)e(come)228 1022 y(up)e(with)f(something)g (more)g(useful)h(than)g(a)f Fr(PSP)m(A)o(CE)p Fx(-completeness)f (classi\014cation.)228 1076 y(\(3\))18 b(Suggesting)h(that)g(the)g (usual)h(framew)o(ork)e(for)g(concrete)i(complexit)o(y)e(analysis)228 1130 y(is)j(wrong-headed,)i(historically)c(con)o(tingen)o(t,)j (unnatural)f(\(esp)q(ecially)f(w)o(orst-case)228 1184 y(asymptotic)15 b(analysis\),)g(and)i(re\015ects)g(an)g(unhapp)o(y)g (state)f(of)h(in)o(teraction)e(b)q(et)o(w)o(een)228 1238 y(computer)g(science)h(theory)f(and)g(practice.)228 1307 y(Regarding)10 b(the)g(third)g(p)q(oin)o(t,)g(most)f(theorists)g(ha)o (v)o(e)h(probably)g(heard)g(similar)e(c)o(harges)228 1361 y(and)13 b(complain)o(ts)f(from)g(colleagues)g(in)h(applied)g (areas)f(of)h(computer)g(science)h(in)f(their)228 1415 y(o)o(wn)h(departmen)o(ts.)228 1523 y Fs(Example)23 b(2:)34 b(An)23 b(Encoun)o(ter)h(With)f(a)h(Computational)f(Biologist.)35 b Fx(In)228 1577 y(recen)o(t)11 b(con)o(v)o(ersations)e(with)h(a)h (biologist)d(who)j(is)f(hea)o(vily)g(in)o(v)o(olv)o(ed)g(in)h(com)o (binatorial)228 1630 y(computing)f([)p Fs(F)l(el97)o Fx(],)g(the)h(follo)o(wing)d(summary)i(w)o(as)g(o\013ered)g(of)g(his)g (in)o(teraction)f(with)228 1684 y(theoretical)14 b(computer)h(scien)o (tists.)359 1758 y Fr(\\A)o(b)n(out)21 b(ten)f(ye)n(ars)h(ago)g(some)f (c)n(omputer)i(scientists)d(c)n(ame)i(by)f(and)359 1812 y(said)11 b(they)h(he)n(ar)n(d)g(we)g(have)g(some)g(r)n(e)n(al)r(ly)f (c)n(o)n(ol)g(pr)n(oblems.)19 b(They)11 b(showe)n(d)359 1866 y(that)17 b(the)f(pr)n(oblems)g(ar)n(e)g Fx(NP)p Fr(-c)n(omplete)g(and)h(went)f(away!")228 1940 y Fx(If)g(this)f(in)o (teraction)g(had)h(b)q(een)h(more)e(recen)o(t,)h(then)g(p)q(erhaps)h (the)f(computer)g(scien-)228 1994 y(tists)d(w)o(ould)g(also)h(ha)o(v)o (e)f(pro)o(v)o(ed)h(that)g(the)g(problems)g(are)g(unlik)o(ely)f(to)h (ha)o(v)o(e)f(e\016cien)o(t)228 2048 y(appro)o(ximation)g(algorithms.) 228 2156 y Fs(The)g(P)o(erv)m(asiv)o(e)c(Nature)k(of)f(Computational)h (In)o(tractabilit)o(y)e(and)i(V)l(arious)228 2210 y(Coping)g (Strategies.)19 b Fx(Naturally)l(,)10 b(man)o(y)i(computer)g(science)g (practitioners)f(w)o(ould)228 2264 y(lik)o(e)k(to)h(sho)q(ot)f(the)i (messenger)f(who)g(brings)g(so)g(m)o(uc)o(h)g(bad)g(news!)23 b(In)17 b(this)f(di\016cult)228 2318 y(situation,)g(computer)i(science) g(theory)f(has)h(articulated)e(a)i(few)f(general)g(programs)228 2372 y(for)e(systematically)e(coping)i(with)g(the)h(ubiquitous)f (phenomena)h(of)f(computational)228 2426 y(in)o(tractabilit)o(y)l(.)25 b(W)l(e)19 b(list)d(these)j(famous)e(basic)h(approac)o(hes,)g(and)g (add)g(to)g(this)f(list)228 2480 y(our)e(less)g(w)o(ell)f(kno)o(wn)h (candidate.)315 2554 y Fl(\017)21 b Fx(The)11 b(idea)g(of)g(fo)q (cusing)g(on)g(a)o(v)o(erage-case)g(as)g(opp)q(osed)h(to)e(w)o (orst-case)g(analysis)359 2608 y(of)k(problems.)p eop %%Page: 3 3 3 2 bop 315 213 a Fl(\017)21 b Fx(The)f(idea)g(of)g(settling)e(for)i (appro)o(ximate)e(solutions)h(to)h(problems,)g(and)h(of)359 267 y(lo)q(oking)14 b(for)g(e\016cien)o(t)h(appro)o(ximation)e (algorithms.)315 321 y Fl(\017)21 b Fx(The)15 b(idea)g(of)g(using)g (randomization)e(in)i(algorithms.)315 374 y Fl(\017)21 b Fx(The)f(idea)f(of)h(harnessing)f(quan)o(tum)h(mec)o(hanics,)g(or)f (biomolecular)g(c)o(hem-)359 428 y(istry)l(,)24 b(to)e(create)h (qualitativ)o(ely)e(more)i(p)q(o)o(w)o(erful)f(computational)g(mec)o (ha-)359 482 y(nisms.)315 536 y Fl(\017)f Fx(The)14 b(idea)g(of)f (devising)h(\014xed-parameter)g(tractable)f(algorithms)f(for)i(param-) 359 590 y(eterizations)f(of)i(a)g(problem.)228 658 y(A)e(list)f(suc)o (h)i(as)f(this)f(cannot)i(b)q(e)g(considered)f(complete)g(without)f (including)i(another)228 712 y(coping)f(strategy)g(that)g(mathematical) f(theorists)h(ha)o(v)o(e)g(frequen)o(tly)h(con)o(tributed)f(to.)228 766 y(This)j(approac)o(h)g(an)o(tedates)g(mo)q(dern)h(complexit)o (y-theoretic)e(framew)o(orks)g(and)h(has)228 820 y(recen)o(tly)21 b(regained)h(some)f(resp)q(ectabilit)o(y)g(among)g(theorists)f(as)i (re\015ected)g(in)g(the)228 874 y(DIMA)o(CS)15 b(Challenges)g(and)g(in) h(new)f(journals)g(suc)o(h)h(as)f(the)h Fr(Journal)g(of)h(Heuristics) 228 928 y Fx(and)e(the)g(A)o(CM)g Fr(Journal)h(of)h(Exp)n(erimental)e (A)o(lgorithms)p Fx(:)315 995 y Fl(\017)21 b Fx(The)g(design)f(of)h (mathematically)d(informed,)k(but)f(p)q(erhaps)g(unanalyzable)359 1049 y(heuristics,)i(that)e(are)h(empirically)e(ev)m(aluated)i(b)o(y)g (their)g(p)q(erformance)g(on)359 1103 y(sets)15 b(of)f(b)q(enc)o(hmark) i(instances.)228 1171 y(The)f(actual)f(state)g(of)g(the)h(practical)f (w)o(orld)g(of)g(computing)h(is)f(that)g(generally)g(there)228 1225 y(is)i(not)g(m)o(uc)o(h)g(systematic)f(connection)i(to)f(w)o(ork)f (in)h(theoretical)f(computer)i(science)228 1278 y(on)g(algorithms)d (and)j(complexit)o(y)l(.)24 b(It)17 b(is)g(heuristic)f(algorithms)f (that)h(are)h(relied)f(on)228 1332 y(in)f(most)f(applications.)607 1447 y Fs(2.)26 b(Fixed-P)o(arameter)15 b(T)l(ractabilit)o(y)228 1528 y Fx(The)21 b(basic)g(concept)h(of)e(the)h(parameterized)g (complexit)o(y)f(framew)o(ork)g(is)g(that)h(of)228 1582 y Fr(\014xe)n(d-p)n(ar)n(ameter)c(tr)n(actability)p Fx(.)23 b(The)16 b(de\014nition)g(is)g(b)q(est)h(in)o(tro)q(duced)f(through)g (con-)228 1636 y(crete)f(examples.)228 1744 y Fm(Ver)m(tex)j(Co)o(ver) 228 1798 y Fk(Instance:)i Fx(A)c(graph)e Fj(G)f Fx(=)g(\()p Fj(V)s(;)8 b(E)s Fx(\))k(and)j(a)g(p)q(ositiv)o(e)f(in)o(teger)h Fj(k)q Fx(.)228 1852 y Fk(P)o(arameter:)j Fj(k)228 1906 y Fk(Question:)k Fx(Do)q(es)16 b Fj(G)h Fx(ha)o(v)o(e)f(a)g(v)o(ertex)g (co)o(v)o(er)g(of)g(size)g Fj(k)q Fx(?)25 b(That)15 b(is,)h(is)g(there) h(a)f(subset)228 1960 y Fj(V)264 1944 y Fi(0)289 1960 y Fl(\022)e Fj(V)25 b Fx(of)16 b(size)f(at)g(most)g Fj(k)i Fx(suc)o(h)f(that)f(for)g(ev)o(ery)g(edge)h Fj(uv)f Fl(2)f Fj(E)s Fx(,)g(either)i Fj(u)d Fl(2)h Fj(V)1654 1944 y Fi(0)1682 1960 y Fx(or)228 2014 y Fj(v)g Fl(2)f Fj(V)344 1998 y Fi(0)355 2014 y Fx(?)228 2095 y Fm(Domina)m(ting)j(Set)228 2149 y Fk(Instance:)k Fx(A)c(graph)e Fj(G)f Fx(=)g(\()p Fj(V)s(;)8 b(E)s Fx(\))k(and)j(a)g(p)q(ositiv)o(e)f(in)o(teger)h Fj(k)q Fx(.)228 2203 y Fk(P)o(arameter:)j Fj(k)228 2257 y Fk(Question:)g Fx(Do)q(es)13 b Fj(G)g Fx(ha)o(v)o(e)f(a)g(dominating) g(set)h(of)f(size)h Fj(k)q Fx(?)20 b(That)12 b(is,)g(is)h(there)f(a)h (subset)228 2311 y Fj(V)264 2294 y Fi(0)290 2311 y Fl(\022)h Fj(V)26 b Fx(of)16 b(size)g(at)f(most)g Fj(k)i Fx(suc)o(h)f(that)g(ev)o (ery)g(v)o(ertex)f Fj(u)f Fl(2)h Fj(V)25 b Fx(of)16 b Fj(G)f Fx(either)h(b)q(elongs)228 2365 y(to)e Fj(V)320 2348 y Fi(0)347 2365 y Fx(or)g(has)i(a)e(neigh)o(b)q(or)h(in)g Fj(V)798 2348 y Fi(0)810 2365 y Fx(?)228 2446 y Fm(The)i(Steiner)g(Pr)o (oblem)i(f)o(or)e(Hyper)o(cubes)228 2500 y Fk(Instance:)28 b Fx(A)19 b(set)g Fj(S)i Fx(=)f Fl(f)p Fj(x)715 2507 y Fh(i)748 2500 y Fx(:)e(1)h Fl(\024)h Fj(i)e Fl(\024)i Fj(k)q Fl(g)f Fx(of)f(binary)h(v)o(ectors,)g Fj(x)1432 2507 y Fh(i)1465 2500 y Fl(2)g(f)p Fx(0)p Fj(;)8 b Fx(1)p Fl(g)1627 2483 y Fh(n)1668 2500 y Fx(for)228 2554 y Fj(i)k Fx(=)h(1)p Fj(;)8 b(:::;)g(k)q Fx(,)k(and)j(a)g(p)q(ositiv)o(e)f(in)o (teger)h Fj(m)p Fx(.)228 2608 y Fk(P)o(arameter:)j Fj(k)p eop %%Page: 4 4 4 3 bop 228 213 a Fk(Question:)19 b Fx(Is)14 b(there)g(a)f(tree)h Fj(T)19 b Fx(=)12 b(\()p Fj(V)s(;)c(E)s Fx(\))j(and)j(a)f(lab)q(eling)g (of)h(the)g(v)o(ertices)f(of)g Fj(T)20 b Fx(with)228 267 y(elemen)o(ts)15 b(of)f Fl(f)p Fx(0)p Fj(;)8 b Fx(1)p Fl(g)578 250 y Fh(n)615 267 y Fx(suc)o(h)16 b(that)e(the)h(follo)o (wing)e(conditions)i(are)g(satis\014ed?)228 321 y(\(1\))f(The)h(lea)o (v)o(es)g(are)g(lab)q(eled)g(1:1)f(with)h(the)g(elemen)o(ts)g(of)g Fj(S)s Fx(.)228 374 y(\(2\))g(The)i(sum)f(o)o(v)o(er)g(the)g(edges)h Fj(uv)g Fx(of)f Fj(T)23 b Fx(of)15 b(the)i(Hamming)e(distance)h(b)q(et) o(w)o(een)h(the)228 428 y(lab)q(els)e Fj(l)q Fx(\()p Fj(u)p Fx(\))c Fl(2)i(f)p Fx(0)p Fj(;)8 b Fx(1)p Fl(g)600 412 y Fh(n)637 428 y Fx(and)15 b Fj(l)q Fx(\()p Fj(v)r Fx(\))c Fl(2)i(f)p Fx(0)p Fj(;)8 b Fx(1)p Fl(g)967 412 y Fh(n)1004 428 y Fx(is)15 b(at)f(most)h Fj(m)p Fx(.)228 522 y Fm(The)i(Maximum)f(A)o(greement)i(Subtree)h(Pr)o(oblem)d Fx(\(MAST\))228 576 y Fk(Instance:)k Fx(A)15 b(set)g(of)g(ro)q(oted)g (trees)g Fj(T)877 583 y Ft(1)896 576 y Fj(;)8 b(:::;)g(T)1004 583 y Fh(r)1035 576 y Fx(\()p Fj(r)13 b Fl(\025)g Fx(3\))h(with)h(the)g (leaf)g(set)g(of)f(eac)o(h)i Fj(T)1709 583 y Fh(i)228 630 y Fx(lab)q(eled)f(1:1)f(with)h(a)g(set)g(of)f(sp)q(ecies)i Fj(X)t Fx(,)e(and)i(a)f(p)q(ositiv)o(e)f(in)o(teger)g Fj(k)q Fx(.)228 684 y Fk(P)o(arameter:)k Fj(k)228 738 y Fk(Question:)i Fx(Is)c(there)f(a)h(subset)f Fj(S)h Fl(\022)d Fj(X)19 b Fx(of)c(size)g(at)g(most)g Fj(k)h Fx(suc)o(h)g(that)f Fj(T)1506 745 y Fh(i)1535 738 y Fx(restricted)228 792 y(to)f(the)h(leaf)f(set)g Fj(X)557 775 y Fi(0)581 792 y Fx(=)f Fj(X)f Fl(\000)e Fj(S)17 b Fx(is)d(the)h(same)f(\(up)h(to) f(lab)q(el-preserving)h(isomorphism)228 846 y(and)g(ignoring)f(v)o (ertices)h(of)f(degree)i(2\))e(for)h Fj(i)d Fx(=)h(1)p Fj(;)8 b(:::;)g(r)q Fx(?)228 954 y(All)21 b(of)h(these)g(problems)g (are)f Fr(NP)p Fx(-complete)h(and)g(they)g(are)g(describ)q(ed)h(ab)q(o) o(v)o(e)e(in)228 1007 y(the)h(standard)g(w)o(a)o(y)f(for)h(the)h (parameterized)e(complexit)o(y)h(framew)o(ork.)39 b(P)o(art)22 b(of)228 1061 y(the)e(input)g(\(whic)o(h)g(ma)o(y)g(b)q(e)h(some)f (aggregate)f(of)g(v)m(arious)h(asp)q(ects\))g(is)g(iden)o(ti\014ed)228 1115 y(as)i(the)h Fr(p)n(ar)n(ameter)g Fx(for)f(the)h(problem)f(sp)q (eci\014cation.)42 b(\(In)23 b(order)g(to)f(consider)g(a)228 1169 y(parameterized)g(problem)g(classically)l(,)g(just)g(ignore)f(the) i(parameter)e(part)h(of)g(the)228 1223 y(sp)q(eci\014cation.)32 b(Note)19 b(that)f(a)h(single)g(classical)f(problem)h(ma)o(y)g(giv)o(e) f(rise)h(to)g(man)o(y)228 1277 y(di\013eren)o(t)e(parameterized)h (problems.\))28 b(All)18 b(of)f(the)i(four)f(problems)f(ab)q(o)o(v)o(e) h(can)g(b)q(e)228 1334 y(solv)o(ed)c(in)h(time)g Fj(O)q Fx(\()p Fj(n)602 1318 y Fh(f)t Ft(\()p Fh(k)q Ft(\))671 1334 y Fx(\))g(b)o(y)g(simple)g(brute)g(force)g(algorithms.)303 1388 y(P)o(apadimitriou)20 b(and)j(Y)l(annak)m(akis)g(sho)o(w)o(ed)g (that)f Fm(Ver)m(tex)k(Co)o(ver)d Fx(can)g(b)q(e)228 1442 y(solv)o(ed)14 b(in)g(time)f Fj(O)q Fx(\(3)595 1426 y Fh(k)616 1442 y Fj(n)p Fx(\))h([)p Fs(PY96)o Fx(].)20 b(Balasubramanian,)12 b(F)l(ello)o(ws)h(and)i(Raman)f(ga)o(v)o(e)228 1497 y(an)e(algorithm)f(with)h(running)h(time)e Fj(O)q Fx(\(\(53)p Fj(=)p Fx(40\))1066 1481 y Fh(k)1085 1497 y Fj(k)1110 1481 y Ft(2)1135 1497 y Fx(+)5 b Fj(k)q(n)p Fx(\))13 b([)p Fs(BFR98)p Fx(].)19 b(It)12 b(is)g(p)q(ossible)228 1551 y(to)h(do)i(ev)o(en)f(b)q(etter.)20 b(Note)14 b(that)f(since)h (the)h(exp)q(onen)o(tial)e(dep)q(endence)k(on)d(the)h(para-)228 1605 y(meter)f Fj(k)h Fx(in)f(the)h(last)e(expression)h(is)g Fr(additive)p Fx(,)h Fm(Ver)m(tex)i(Co)o(ver)d Fx(is)g Fr(wel)r(l-solve)n(d)h(for)228 1659 y(input)f(of)h(any)f(size)f(so)h (long)g(as)g Fj(k)i Fr(is)d(no)h(mor)n(e)h(than)f(ar)n(ound)h(60)p Fx(.)20 b(The)13 b(di\013erence)g(b)q(e-)228 1713 y(t)o(w)o(een)h(the)g (situation)f(for)g Fm(Domina)m(ting)i(Set)f Fx(and)h Fm(Ver)m(tex)i(Co)o(ver)d Fx(is)g(displa)o(y)o(ed)228 1767 y(in)h(T)l(able)g(1.)p 519 1846 898 2 v 518 1899 2 54 v 691 1899 V 744 1883 a Fj(n)e Fx(=)g(50)p 929 1899 V 92 w Fj(n)g Fx(=)g(100)p 1167 1899 V 87 w Fj(n)g Fx(=)g(150)p 1415 1899 V 519 1901 898 2 v 518 1955 2 54 v 551 1939 a Fj(k)h Fx(=)f(2)p 691 1955 V 116 w(625)p 929 1955 V 152 w(2,500)p 1167 1955 V 138 w(5,625)p 1415 1955 V 519 1957 898 2 v 518 2011 2 54 v 551 1995 a Fj(k)h Fx(=)f(3)p 691 2011 V 87 w(15,625)p 929 2011 V 99 w(125,000)p 1167 2011 V 92 w(421,875)p 1415 2011 V 519 2012 898 2 v 518 2066 2 54 v 551 2050 a Fj(k)h Fx(=)f(5)p 691 2066 V 76 w(390,625)p 929 2066 V 69 w(6,250,000)p 1167 2066 V 45 w(31,640,625)p 1415 2066 V 519 2068 898 2 v 518 2122 2 55 v 540 2106 a Fj(k)g Fx(=)g(10)p 691 2122 V 41 w(1)p Fj(:)p Fx(9)c Fl(\002)i Fx(10)872 2089 y Ft(12)p 929 2122 V 950 2106 a Fx(9)p Fj(:)p Fx(8)e Fl(\002)i Fx(10)1110 2089 y Ft(14)p 1167 2122 V 1193 2106 a Fx(3)p Fj(:)p Fx(7)f Fl(\002)g Fx(10)1353 2089 y Ft(16)p 1415 2122 V 519 2124 898 2 v 518 2178 2 55 v 540 2162 a Fj(k)j Fx(=)g(20)p 691 2178 V 41 w(1)p Fj(:)p Fx(8)c Fl(\002)i Fx(10)872 2145 y Ft(26)p 929 2178 V 950 2162 a Fx(9)p Fj(:)p Fx(5)e Fl(\002)i Fx(10)1110 2145 y Ft(31)p 1167 2178 V 1193 2162 a Fx(2)p Fj(:)p Fx(1)f Fl(\002)g Fx(10)1353 2145 y Ft(35)p 1415 2178 V 519 2180 898 2 v 408 2247 a Fm(T)l(able)17 b(1.)23 b Fx(The)15 b(Ratio)845 2229 y Fh(n)866 2218 y Fg(k)q Fq(+1)p 845 2236 79 2 v 855 2265 a Ft(2)873 2256 y Fg(k)892 2265 y Fh(n)944 2247 y Fx(for)f(V)l(arious)h(V)l(alues)g(of)g Fj(n)g Fx(and)h Fj(k)q Fx(.)228 2446 y(There)d(are)g(m)o(yriad)f(w)o(a)o(ys)g(in)g (whic)o(h)h(n)o(um)o(b)q(ers)h(that)e(are)h(small)e(or)i(mo)q(derately) f(large)228 2500 y(\(e.g.,)i Fj(k)g Fl(\024)f Fx(40\))i(arise)g (naturally)f(in)h(problem)g(sp)q(eci\014cations.)21 b(F)l(or)14 b(example,)h(graph)228 2554 y(linear)e(la)o(y)o(out)g(width)g(metrics)h (are)g(of)f(in)o(terest)h(in)g(VLSI)h(la)o(y)o(out)e(and)h(routing)f (prob-)228 2608 y(lems)j(and)i(ha)o(v)o(e)e(imp)q(ortan)o(t)g (applications)f(for)i(width)f(v)m(alues)h(of)g Fj(k)g Fl(\024)f Fx(10.)24 b(In)o(terv)m(al)p eop %%Page: 5 5 5 4 bop 228 213 a Fx(graphs)14 b(of)g(path)o(width)g Fj(k)g Fl(\024)e Fx(10)i(ha)o(v)o(e)h(applications)e(in)h(DNA)g (sequence)i(reconstruc-)228 267 y(tion)i(problems)h([)p Fs(BDFHW95)p Fx(].)31 b Fr(L)n(o)n(gic)19 b(and)g(datab)n(ase)h(pr)n (oblems)f Fx(frequen)o(tly)f(are)228 321 y(de\014ned)g(as)e(ha)o(ving)g (input)h(consisting)f(a)g(form)o(ula)f(\(whic)o(h)i(ma)o(y)f(b)q(e)h (small)e(and)i(rel-)228 374 y(ativ)o(ely)f(in)o(v)m(arian)o(t\),)g(and) i(some)f(other)h(structure)f(\(suc)o(h)h(as)f(a)g(database\))g(whic)o (h)h(is)228 428 y(t)o(ypically)c(quite)i(large)g(and)g(c)o(hangeable.) 23 b(F)l(orm)o(ula)15 b(size,)h(or)g(other)f(asp)q(ects)i(of)f(for-)228 482 y(m)o(ula)c(structure)g(ma)o(y)h(b)q(e)g(a)g(relev)m(an)o(t)f (parameter)g([)p Fs(Y)l(an95)p Fx(].)18 b Fr(Har)n(dwar)n(e)d(c)n (onstr)n(aints)228 536 y Fx(are)i(a)g(common)g(source)g(of)g(natural)f (parameters.)26 b(The)17 b(n)o(um)o(b)q(er)h(of)f(pro)q(cessors)g(or) 228 590 y(mac)o(hines)e(to)g(b)q(e)i(sc)o(heduled)f(ma)o(y)f(b)q(e)i(b) q(ounded)g(b)o(y)f(a)f(v)m(alue)h(suc)o(h)g(as)f Fj(k)g Fl(\024)f Fx(20.)21 b(The)228 644 y(n)o(um)o(b)q(er)c(of)g(degrees)g (of)g(freedom)f(in)h(a)g(rob)q(ot)f(motion-planning)g(problem)h(is)f (com-)228 698 y(monly)21 b(in)g(the)g(range)g Fj(k)j Fl(\024)f Fx(10.)38 b(The)21 b(n)o(um)o(b)q(er)h(of)f(wiring)f(la)o(y)o (ers)g(in)h(VLSI)h(c)o(hip)228 752 y(man)o(ufacture)16 b(is)h(t)o(ypically)e(b)q(ounded)j(b)o(y)f Fj(k)g Fl(\024)e Fx(30.)25 b(Imp)q(ortan)o(t)16 b(distributional)f(pa-)228 806 y(rameters)j(ma)o(y)h(also)f(arise)g(in)h(w)o(a)o(ys)f(that)h(are)g (not)f(at)h(all)f(ob)o(vious.)31 b(Thorup,)20 b(for)228 860 y(example,)h(has)g(sho)o(wn)f(that)g(the)h(\015o)o(w)f(graphs)h(of) f(structured)h(programs)e(for)h(the)228 914 y(ma)s(jor)14 b(computer)h(languages)f(ha)o(v)o(e)h(treewidth)f Fj(k)g Fl(\024)f Fx(7)i([)p Fs(Th97)p Fx(].)228 1022 y Fs(The)f(Basic)g (De\014nitions.)k Fx(The)12 b(basic)g(de\014nitions)g(of)g (parameterized)f(complexit)o(y)228 1076 y(are)k(as)f(follo)o(ws.)228 1151 y Fs(De\014nition.)32 b Fx(A)20 b Fr(p)n(ar)n(ameterize)n(d)g (language)f Fj(L)g Fx(is)g(a)g(subset)h Fj(L)f Fl(\022)i Fx(\006)1463 1135 y Fi(\003)1495 1151 y Fl(\002)13 b Fx(\006)1576 1135 y Fi(\003)1596 1151 y Fx(.)33 b(If)19 b Fj(L)228 1205 y Fx(is)f(a)g(parameterized)g(language)g(and)g(\()p Fj(x;)8 b(y)r Fx(\))17 b Fl(2)h Fj(L)g Fx(then)h(w)o(e)f(will)f(refer)i (to)f Fj(x)g Fx(as)g(the)228 1259 y Fr(main)h(p)n(art)p Fx(,)g(and)f(refer)g(to)g Fj(y)i Fx(as)e(the)g Fr(p)n(ar)n(ameter)p Fx(.)29 b(It)18 b(mak)o(es)g(no)g(di\013erence)h(to)e(the)228 1313 y(theory)e(and)g(is)g(o)q(ccasionally)e(more)i(con)o(v)o(enien)o (t)g(to)f(consider)h(that)g Fj(y)i Fx(is)d(an)h(in)o(teger,)228 1367 y(or)d(equiv)m(alen)o(tly)g(to)g(de\014ne)i(a)e(parameterized)g (language)g(to)g(b)q(e)h(a)g(subset)g(of)f(\006)1595 1351 y Fi(\003)1619 1367 y Fl(\002)5 b Fj(I)-12 b(N)5 b Fx(.)228 1486 y Fs(De\014nition.)28 b Fx(A)19 b(parameterized)e (language)h Fj(L)g Fx(is)g Fr(\014xe)n(d-p)n(ar)n(ameter)h(tr)n (actable)f Fx(if)g(it)228 1540 y(can)i(b)q(e)g(determined)g(in)g(time)f Fj(f)5 b Fx(\()p Fj(k)q Fx(\))p Fj(n)905 1523 y Fh(\013)949 1540 y Fx(whether)20 b(\()p Fj(x;)8 b(k)q Fx(\))19 b Fl(2)h Fj(L)p Fx(,)g(where)g Fl(j)p Fj(x)p Fl(j)g Fx(=)g Fj(n)p Fx(,)h Fj(\013)228 1594 y Fx(is)c(a)g(constan)o(t)f(indep)q (enden)o(t)j(of)e(b)q(oth)h Fj(n)f Fx(and)h Fj(k)g Fx(and)g Fj(f)23 b Fx(is)17 b(an)g(arbitrary)f(function.)228 1648 y(The)d(family)g(of)g(\014xed-parameter)g(tractable)f(parameterized)h (languages)g(is)g(denoted)228 1702 y Fr(FPT)p Fx(.)228 1777 y(It)20 b(is)g(somewhat)f(surprising,)h(although)f(the)h(argumen)o (t)g(is)f(not)h(hard,)h(that)e Fr(FPT)228 1831 y Fx(is)j(unc)o(hanged)h (if)f(the)h(de\014nition)f(ab)q(o)o(v)o(e)g(is)g(mo)q(di\014ed)h(b)o(y) f(replacing)g Fj(f)5 b Fx(\()p Fj(k)q Fx(\))p Fj(n)1627 1814 y Fh(\013)1674 1831 y Fx(b)o(y)228 1885 y Fj(f)g Fx(\()p Fj(k)q Fx(\))k(+)i Fj(n)398 1868 y Fh(\013)438 1885 y Fx([)p Fs(CCDF97)p Fx(].)303 1939 y(\\Ab)q(out)i(half)t(")f(of)i (the)f(naturally)f(parameterized)i(problems)f(cataloged)f(as)h Fr(NP)p Fx(-)228 1993 y(complete)19 b(in)g(the)g(b)q(o)q(ok)g(b)o(y)g (Garey)g(and)g(Johnson)h([)p Fs(GJ79)p Fx(])e(are)h(in)g Fr(FPT)p Fx(,)f(includ-)228 2047 y(ing)f(three)h(of)f(the)g(six)g (basic)h(problems)f(singled)g(out)g(for)g(atten)o(tion)f(in)i(Chapter)f (3.)228 2101 y(It)j(is)g(alw)o(a)o(ys)f(p)q(ossible)i(to)e (parameterize)h(a)h(problem)f(in)g(v)m(arious)g(w)o(a)o(ys)g(that)f (are)228 2155 y(\014xed-parameter)c(tractable,)f(y)o(et)h(it)g(is)g (not)g(surprising)f(that)h(man)o(y)g(parameterized)228 2209 y(problems)k(apparen)o(tly)f(do)h(not)g(b)q(elong)h(to)e Fr(FPT)p Fx(.)g(The)i(naturally)e(parameterized)228 2263 y Fm(Domina)m(ting)f(Set)f Fx(problem)g(de\014ned)i(ab)q(o)o(v)o(e)e (is)g(one)g(of)g(these.)23 b(W)l(e)17 b(can)f(\014nd)h(evi-)228 2317 y(dence)g(for)e(\014xed-parameter)h Fr(intr)n(actability)g Fx(b)o(y)f(studying)h(the)g(appropriate)f(notion)228 2371 y(of)g(problem)f(transformation.)228 2446 y Fs(De\014nition.)19 b Fx(A)13 b Fr(p)n(ar)n(ametric)i(tr)n(ansformation)e Fx(from)f(a)h(parameterized)g(language)f Fj(L)228 2500 y Fx(to)17 b(a)g(parameterized)h(language)e Fj(L)847 2483 y Fi(0)877 2500 y Fx(is)h(an)h(algorithm)d(that)i(computes)h(from) f(input)228 2554 y(consisting)d(of)h(a)g(pair)f(\()p Fj(x;)8 b(k)q Fx(\),)14 b(a)g(pair)h(\()p Fj(x)931 2537 y Fi(0)942 2554 y Fj(;)8 b(k)988 2537 y Fi(0)999 2554 y Fx(\))15 b(suc)o(h)h(that:)228 2608 y(\(1\))e(\()p Fj(x;)8 b(k)q Fx(\))j Fl(2)i Fj(L)i Fx(if)g(and)g(only)g(if)g(\()p Fj(x)825 2591 y Fi(0)836 2608 y Fj(;)8 b(k)882 2591 y Fi(0)893 2608 y Fx(\))k Fl(2)h Fj(L)997 2591 y Fi(0)1009 2608 y Fx(,)p eop %%Page: 6 6 6 5 bop 228 213 a Fx(\(2\))14 b Fj(k)326 196 y Fi(0)350 213 y Fx(=)f Fj(g)r Fx(\()p Fj(k)q Fx(\))h(is)h(a)g(function)g(only)g (of)f Fj(k)q Fx(,)h(and)228 267 y(\(3\))g(the)i(computation)e(is)h (accomplished)g(in)g(time)g Fj(f)5 b Fx(\()p Fj(k)q Fx(\))p Fj(n)1248 250 y Fh(\013)1273 267 y Fx(,)16 b(where)h Fj(n)e Fx(=)f Fl(j)p Fj(x)p Fl(j)p Fx(,)i Fj(\013)h Fx(is)e(a)228 321 y(constan)o(t)f(indep)q(enden)o(t)j(of)d(b)q(oth)i Fj(n)f Fx(and)h Fj(k)q Fx(,)e(and)i Fj(f)k Fx(is)15 b(an)g(arbitrary)f (function.)228 386 y Fs(An)24 b(Illustrativ)o(e)e(Non-Example.)37 b Fx(It)21 b(can)h(b)q(e)g(helpful)f(to)f(observ)o(e)h(ho)o(w)g(pa-)228 440 y(rametric)c(transformations)f(di\013er)i(from)g(ordinary)g(p)q (olynomial-time)f(reductions.)228 494 y(Recall)d(that)g(for)g(a)g (graph)g Fj(G)f Fx(=)g(\()p Fj(V)s(;)8 b(E)s Fx(\))j(on)k Fj(n)f Fx(v)o(ertices,)g(a)g(set)h(of)f(v)o(ertices)g Fj(V)1568 478 y Fi(0)1593 494 y Fl(\022)f Fj(V)24 b Fx(is)228 548 y(a)14 b Fj(k)q Fx(-clique)g(in)g Fj(G)g Fx(if)g(and)g(only)g(if)g Fj(V)k Fl(\000)8 b Fj(V)929 532 y Fi(0)955 548 y Fx(is)13 b(a)h(v)o(ertex)g(co)o(v)o(er)g(in)g(the)g(complemen)o(tary)228 602 y(graph)c Fj(G)388 586 y Fi(0)411 602 y Fx(where)h(v)o(ertices)f 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y(of)j(represen)o(tativ)o(e)g(b)q(enc)o(hmarks.)26 b(The)17 b(idea)f(that)h(a)o(v)o(erage-case)f(analysis)f(is)i(more)228 2392 y(realistic)g(than)h(w)o(orst-case)g(analysis)f(has)h(b)q(een)i (around)f(since)f(the)h(b)q(eginnings)g(of)228 2446 y(theoretical)13 b(computer)i(science.)20 b(Although)15 b(w)o(e)f(ha)o(v)o(e)g(a)h(few)f (gems)h(of)f(a)o(v)o(erage-case)228 2500 y(analysis,)j(in)h(general)g (the)g(program)f(seems)h(to)g(b)q(e)h(to)q(o)e(di\016cult)h(to)f(carry) h(out)g(for)228 2554 y(t)o(ypical)d(hard)i(problems,)g(realistic)e (distributions)h(and)h(sophisticated)f(algorithms.)228 2608 y(It)d(is)f(also)f(frequen)o(tly)i(unclear)f(what)g(constitutes)g (a)g(reasonable)g(assumption)g(ab)q(out)p eop %%Page: 9 9 9 8 bop 228 213 a Fx(the)17 b(distribution)e(of)h(problem)h(instances,) f(apart)g(from)g(the)h(analytical)e(di\016culties)228 267 y(of)g(the)g(program.)228 321 y(A)h(completeness)g(notion)f(for)g (a)o(v)o(erage-case)g(complexit)o(y)g(has)h(b)q(een)h(in)o(tro)q(duced) f(b)o(y)228 374 y(Levin)h([)p Fs(Lev86)o Fx(].)23 b(This)16 b(also)f(seems)i(to)e(b)q(e)i(di\016cult)f(to)f(apply)l(,)i(and)f(has)g (only)g(b)q(een)228 428 y(demonstrated)21 b(for)g(a)g(few)h(problems.) 39 b(The)21 b(main)g(w)o(eaknesses)h(of)f(a)o(v)o(erage-case)228 482 y(analysis)16 b(as)g(a)h(mathematical)f(program)g(for)g(coping)h (with)f(in)o(tractabilit)o(y)e(seem)k(to)228 536 y(b)q(e:)315 606 y Fl(\017)j Fx(In)h(general,)g(it)f(seems)h(to)f(b)q(e)h(to)q(o)f (di\016cult)g(to)g(pro)o(v)o(e)g(the)h(mathematical)359 660 y(results)14 b(that)h(the)g(program)f(calls)g(for.)1051 644 y Ft(5)315 714 y Fl(\017)21 b Fx(The)f(p)q(ositiv)o(e)f(to)q(olkit) f(has)i(not)g(dev)o(elop)q(ed)h(w)o(ell.)33 b(F)l(ew)20 b(general)g(metho)q(ds)359 768 y(are)15 b(kno)o(wn)h(for)f(designing)g (algorithms)f(with)h(pro)o(v)m(ably)g(go)q(o)q(d)h(a)o(v)o(erage-case) 359 822 y(p)q(erformance.)j(\(One)14 b(in)o(teresting)e(example)h(of)g (a)g(general)g(metho)q(d)h(has)f(b)q(een)359 876 y(describ)q(ed)j(b)o 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y(is)e(at)f(issue)h (in)g(parameterized)g(complexit)o(y)l(.)303 1065 y(A)k(w)o(eakness)g (of)g(the)h(parameterized)e(complexit)o(y)h(program)f(is)h(that)g(some) g(of)228 1119 y(the)e(most)f(general)g(and)h(sp)q(ectacular)g(p)q (ositiv)o(e)f(metho)q(ds,)h(suc)o(h)g(as)g(the)g(celebrated)228 1173 y(results)i(of)h(Rob)q(ertson)g(and)g(Seymour)g([)p Fs(RS85)p Fx(],)g(yield)g(algorithms)d(ha)o(ving)j(para-)228 1227 y(meter)14 b(functions)g(that)g(are)g(supremely)g(impractical)f (\(e.g.,)g(to)o(w)o(ers)g(of)h(2's)g(of)g(heigh)o(t)228 1281 y(describ)q(ed)f(b)o(y)f(to)o(w)o(ers)e(of)i(2's)f(...\).)17 b(If)12 b Fr(tr)n(actability)g Fx(has)g(friends)g(lik)o(e)f(these,)i (who)e(needs)228 1335 y(enemies?)303 1389 y(The)18 b(main)f(strengths)g (of)h(parameterization)d(as)j(a)f(program)g(are)h(that)f(it)g(do)q(es) 228 1443 y(seem)e(to)f(b)q(e)i(v)o(ery)e(generally)g(applicable)h(to)f (hard)h(problems)f(throughout)h(the)g(clas-)228 1497 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y(for)f(coping)h(with)g(in)o(tractabilit)o(y)d(is)j(that)f(it)g(is)h (not)g(a)g(coheren)o(t)g(mathematical)e(pro-)228 2554 y(gram.)24 b(The)17 b(p)q(ositiv)o(e)f(to)q(olkit)f(prop)q(erly)h (includes)i(v)o(o)q(o)q(do)q(o)e(and)h(the)g(kitc)o(hen)g(sink.)228 2608 y(As)d(a)g(program,)f(it)h(do)q(esn't)g(call)f(for)h(an)o(y)g (theorems,)f(only)h(empirical)f(p)q(erformance.)p eop %%Page: 12 12 12 11 bop 228 213 a Fx(The)15 b(undeniable)h(successes)g(of)f (sometimes)g(\\mindless")f(and)i(generally)e(unanalyz-)228 267 y(able)g(heuristic)f(algorithms)f(puts)j(computer)f(science)g (theory)g(in)g(an)g(uncomfortable)228 321 y(p)q(osition.)228 374 y(Heuristics)19 b(based)h(on)f(lo)q(cal)g(searc)o(h)h(p)q(erhaps)g (come)g(the)g(closest)f(to)g(constituting)228 428 y(a)d(mathematically) e(articulated)h(general)g(coping)h(strategy)f(for)h(in)o(tractabilit)o (y)d(\(see)228 482 y(the)j(articles)f(in)i([)p Fs(AL97)o Fx(]\).)23 b(There)17 b(is)e(a)h(negativ)o(e)g(to)q(olkit)e(\(of)i (sorts\))f(based)i(on)f(the)228 536 y(notion)21 b(of)g Fr(p)n(olynomial)i(lo)n(c)n(al)e(se)n(ar)n(ch)g Fx(problems)h(and)g Fr(PLS)p Fx(-completeness,)g(in)o(tro-)228 590 y(duced)f(b)o(y)e (Johnson,)i(P)o(apadimitriou)c(and)j(Y)l(annak)m(akis)g([)p Fs(JPY88)o Fx(].)32 b(Although)19 b(a)228 644 y(n)o(um)o(b)q(er)f(of)g (lo)q(cal)f(searc)o(h)g(problems)h(ha)o(v)o(e)f(b)q(een)j(sho)o(wn)d (to)g(b)q(e)i(complete,)f(the)g(re-)228 698 y(ductions)c(are)h(quite)f (demanding)h(\(so)f(there)g(aren't)g(v)o(ery)g(man)o(y)h(suc)o(h)g (results\),)e(and)228 752 y(there)h(is)g(furthermore)f(a)h(notable)g(p) q(eculiarit)o(y)f(of)h(this)f(framew)o(ork.)18 b(F)l(or)c(a)g(concrete) 228 806 y(example,)f(b)q(ecause)g(the)g Fm(Tra)l(veling)i(Salesman)g (Pr)o(oblem)f Fx(is)e Fr(NP)p Fx(-hard,)h(one)g(re-)228 860 y(sorts)f(to)h(a)h(lo)q(cal)e(searc)o(h)i(heuristic)f(based)h(on)f (the)h Fj(k)q Fx(-Opt)g(neigh)o(b)q(orho)q(o)q(d)g(structure,)228 914 y(suc)o(h)20 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Fx(harder)h(to)e(crac)o(k) h(than)g(w)o(as)f(hop)q(ed)i(in)f(the)h(early)e(y)o(ears)h(of)228 2608 y(the)h(\014eld.)p eop %%Page: 13 13 13 12 bop 660 213 a Fs(4.)26 b(Industrial)16 b(Strength)i Fe(FPT)228 294 y Fx(In)c(this)f(section)g(w)o(e)h(sk)o(etc)o(h)f(the)h (reasons)f(wh)o(y)g(w)o(e)h(think)f(that)g(parameterized)g(com-)228 348 y(plexit)o(y)g(o\013ers)g(an)g(inevitable,)g(general,)h (mathematical)e(program)g(for)i(dealing)f(with)228 401 y(computational)g(in)o(tractabilit)o(y)l(.)k(There)f(are)f(t)o(w)o(o)e (main)i(p)q(oin)o(ts)g(to)f(the)i(argumen)o(t.)303 478 y(1.)k(The)h(parameterized)f(complexit)o(y)g(p)q(ersp)q(ectiv)o(e)h (can)g(lead)f(to)g(useful)h(algo-)359 532 y(rithms)14 b(in)h(sev)o(eral)f(di\013eren)o(t)h(w)o(a)o(ys,)e(including:)434 586 y Fl(\017)20 b Fx(Directly)c(and)h(analytically)l(,)e(when)i(the)g (parameter)f(function)h(is)f(rea-)477 640 y(sonable)c(\(i.e.,)g(not)g (to)q(o)h(fast-gro)o(wing\))d(and)j(the)g(parameter)f(describ)q(es)477 694 y(a)j(restriction)f(of)g(the)i(general)e(problem)h(that)g(is)f (still)g(useful.)434 747 y Fl(\017)20 b Fx(Directly)14 b(and)i(empirically)l(,)d(in)j(cases)f(where)g(the)h(analytic)e(b)q (ound)i(on)477 801 y(the)21 b(running)h(time)e(of)h(the)h Fr(FPT)e Fx(algorithm)f(turns)i(out)g(to)g(b)q(e)h(to)q(o)477 855 y(crude)h(or)e(p)q(essimistic,)i(that)e(is,)i(where)f(the)h (algorithm)d(turns)i(out)477 909 y(to)e(b)q(e)i(useful)f(for)g(larger)f (parameter)g(ranges)h(than)g(the)g(parameter)477 963 y(function)15 b(w)o(ould)g(indicate.)434 1017 y Fl(\017)20 b Fx(By)15 b(supplying)g(systematic)e(guidance)i(in)g(the)g(design)g (of)f(heuristic)h(al-)477 1071 y(gorithms)d(in)h(the)h(form)e(of)h (explicit)g(general)g(metho)q(ds)g(based)h(on)f Fr(FPT)477 1125 y Fx(tec)o(hniques,)h(using)h(the)f(theory)g(to)g(understand)h (\\the)f(in)o(ternal)f(struc-)477 1179 y(ture)22 b(of)g(the)h (complexit)o(y)e(of)h(the)h(problem")f(b)o(y)g(iden)o(tifying)f(those) 477 1233 y(parameterizations)13 b(of)h(the)h(problem)f(that)g(are)h Fr(FPT)e Fx(and)i(those)g(that)477 1287 y(probably)20 b(are)h(not.)36 b(\(E.g.,)21 b(lo)q(cal)f(searc)o(h)g(algorithms)f (that)h(iterate)477 1341 y Fr(FPT)c Fj(k)q Fx(-step)i(explorations)e (of)h(the)h(neigh)o(b)q(orho)q(o)q(d)g(structure)f(of)g(the)477 1395 y(solution)d(space.\))434 1449 y Fl(\017)20 b Fx(Via)e(metho)q (dological)g(connections)h(b)q(et)o(w)o(een)g Fr(FPT)f Fx(and)h(the)g(design)477 1503 y(of)14 b(e\016cien)o(t)h(p)q (olynomial-time)e(appro)o(ximation)g(sc)o(hemes)i(\(where)g(the)477 1557 y(relev)m(an)o(t)j(implicit)f(parameter)h(is)g Fj(k)h Fx(=)g(1)p Fj(=\017)p Fx(,)g(for)e(appro)o(ximations)g(to)477 1611 y(within)e(a)g(factor)g(of)g(\(1)10 b(+)h Fj(\017)p Fx(\))16 b(of)f(optimal\))f(and)i(other)f(appro)o(ximation)477 1665 y(heuristics.)303 1719 y(2.)20 b(The)12 b(notion)g(of)g Fr(FPT)p Fx(,)f(in)h(man)o(y)g(cases,)g(simply)g(pro)o(vides)g(a)g(new) h(name)f(and)g(a)359 1773 y(new)e(p)q(ersp)q(ectiv)o(e)i(on)e (heuristic)g(algorithms)f Fr(alr)n(e)n(ady)j(in)f(use)p Fx(.)18 b(Where)11 b(natural)359 1827 y(instance)20 b(distributions)e (exhibit)i(limited)f(parameter)h(ranges,)g(these)h(ha)o(v)o(e)359 1881 y(frequen)o(tly)16 b(b)q(een)j(exploited)e(in)g(the)g(design)g(of) g(useful)g(heuristics)g(in)g(a)g(w)o(a)o(y)359 1935 y(that)d(the)h (notion)g(of)g Fr(FPT)f Fx(names)h(and)g(systematizes.)228 2011 y(W)l(e)20 b(ha)o(v)o(e)g(already)f(rep)q(orted)h(on)g(a)g (steadily)f(gro)o(wing)g(list)g(of)h(examples)f(of)h Fr(FPT)228 2065 y Fx(problems)11 b(with)h(\\reasonable")f(parametric)f (costs,)i(suc)o(h)g(as)g(2)1308 2048 y Fh(k)1329 2065 y Fx(.)19 b(Curren)o(tly)11 b(the)h(b)q(est)228 2119 y(kno)o(wn)e(algorithm)e(for)h Fm(Ver)m(tex)k(Co)o(ver)d Fx(runs)h(in)f(time)f Fj(O)q Fx(\()p Fj(k)q(n)h Fx(+)h(max)o Fl(f)p Fx(\(1)p Fj(:)p Fx(25542\))1665 2102 y Fh(k)1684 2119 y Fj(k)1709 2102 y Ft(2)1728 2119 y Fx(,)228 2174 y(\(1)p Fj(:)p Fx(2906\))392 2157 y Fh(k)411 2174 y Fx(2)p Fj(:)p Fx(5)p Fj(k)q Fl(g)p Fx(\))16 b(whic)o(h)i(directly)f(indicates) g(that)g(it)g(is)h(useful)f(for)h(graphs)f(of)g(an)o(y)228 2229 y(size,)i(so)g(long)f(as)g Fj(k)i Fl(\024)f Fx(157)f(\(the)h (running)g(time)f(of)h Fj(O)q Fx(\()p Fj(k)q(n)g Fx(+)g(max)o Fl(f)p Fx(\(1)p Fj(:)p Fx(25542\))1646 2212 y Fh(k)1665 2229 y Fj(k)1690 2212 y Ft(2)1710 2229 y Fx(,)228 2284 y(\(1)p Fj(:)p Fx(2906\))392 2268 y Fh(k)411 2284 y Fx(2)p Fj(:)p Fx(5)p Fj(k)q Fl(g)p Fx(\))i(is)g(a)h(result)g(mainly)g(of)g(a)g (com)o(bination)f(of)g(the)i(algorithm)d(b)o(y)228 2338 y(Niedermeier)15 b(and)g(Rossmanith[)p Fs(NR99)o Fx(],)g(the)g (algorithm)e(presen)o(ted)i(in)g([)p Fs(DFS99)p Fx(],)228 2392 y(and)h(a)f(more)h(careful)f(analysis)g([)p Fs(Ste99,)j(SF99)p Fx(]\).)j(In)16 b(fact,)f(exp)q(erimen)o(ts)h(with)f(the)228 2446 y(algorithm)e(indicate)i(that)f(it)h(is)f(useful)i(for)e Fj(k)i Fx(at)f(least)f(up)i(to)f(200)f([)p Fs(HGS98)p Fx(].)303 2500 y Fr(FPT)p Fx(-algorithm)j(design)j(has)g(a)g (distinctiv)o(e)f(to)q(olkit)f(of)i(p)q(ositiv)o(e)f(tec)o(hniques,)228 2554 y(including)d(b)q(ounded)h(treewidth)f(and)g(path)o(width)g(metho) q(ds)g([)p Fs(Bo)q(d96)q Fx(],)f(w)o(ell)h(quasi-)228 2608 y(ordering)c([)p Fs(RS85)p Fx(],)g(color-co)q(ding)g([)p Fs(A)l(YZ94)n Fx(])g(and)h(imp)q(ortan)o(t)e(elemen)o(tary)h(metho)q (ds)p eop %%Page: 14 14 14 13 bop 228 213 a Fx(suc)o(h)20 b(as)f(searc)o(h)h(trees)f(and)h (reduction)g(to)f(a)h(problem)f(k)o(ernel.)34 b(This)19 b(is)g(unfortu-)228 267 y(nately)13 b(not)h(the)g(place)h(to)e(exp)q (osit)h(this)g(in)o(teresting)e(collection)h(of)h(metho)q(ds.)20 b(What)228 321 y(is)13 b(imp)q(ortan)o(t)f(to)i(realize,)f(ho)o(w)o(ev) o(er,)g(is)g(that)g(all)g(of)g(these)h(metho)q(ds)g(can)g(b)q(e)g (system-)228 374 y(atically)c(adapted)i(in)g(the)h(design)f(of)f (heuristic)h(algorithms.)17 b(The)12 b(essen)o(tial)f(theme)i(is)228 428 y(to)h(obtain)g(heuristics)h(from)f Fr(FPT)g Fx(algorithms)e(b)o(y) j(strategies)f(for)g Fr(de\015ating)i(the)g(p)n(a-)228 482 y(r)n(ametric)h(c)n(osts)d Fx(b)o(y)i(truncating)f(or)g(sampling)f (the)i(searc)o(h)f(trees,)g(obstruction)g(sets,)228 536 y(test)f(sets,)h(etc.)20 b(This)15 b(is)f(summarized)h(in)g(the)h (follo)o(wing)c(table.)p 228 592 1505 2 v 227 646 2 54 v 385 630 a Fs(FPT)17 b(T)l(ec)o(hnique)p 894 646 V 279 w(Heuristic)f(Design)g(Strategy)p 1732 646 V 228 648 1505 2 v 227 702 2 54 v 249 685 a Fx(Reduction)f(to)g(a)g(Problem)f (Kernel)p 894 702 V 97 w(A)h(useful)h(pre-pro)q(cessing)f(subroutine)p 1732 702 V 227 756 V 894 756 V 1145 739 a(for)f(an)o(y)h(heuristic.)p 1732 756 V 228 757 1505 2 v 227 811 2 54 v 445 795 a(Searc)o(h)g(T)l (ree)p 894 811 V 238 w(Explore)g(only)g(an)g(a\013ordable,)e (heuristically)p 1732 811 V 227 865 V 894 865 V 1162 849 a(c)o(hosen)j(subtree.)p 1732 865 V 228 867 1505 2 v 227 921 2 54 v 370 905 a(W)l(ell-Quasiordering)p 894 921 V 203 w(Use)f(a)g(sample)g(of)g(the)g(obstruction)f(set.)p 1732 921 V 228 922 1505 2 v 227 976 2 54 v 428 960 a(Color-Co)q(ding)p 894 976 V 264 w(Use)i(a)f(sample)f(of)h(the)g(hash)h(functions.)p 1732 976 V 228 978 1505 2 v 227 1032 2 54 v 365 1016 a(Bounded)g(T)l(reewidth)p 894 1032 V 323 w(Use)f(Angluin's)g(Theorem)p 1732 1032 V 227 1086 V 894 1086 V 1031 1070 a(and)g(a)g(sample)g(of)g (the)g(test)g(set.)p 1732 1086 V 228 1088 1505 2 v 421 1162 a Fm(T)l(able)j(2.)k Fx(Some)16 b Fr(FPT)e Fx(Metho)q(ds)h(and)g (Heuristic)g(Strategies)228 1311 y(F)l(or)j(a)h(sp)q(eci\014c)h (example,)f(the)h Fm(Ga)m(te)h(Ma)m(trix)g(La)m(yout)d Fx(problem)h(is)f(amenable)228 1365 y(to)k(w)o(ell-quasiordering)f (metho)q(ds.)42 b(F)l(or)22 b(width)h(3,)g(there)g(are)g(110)e(minor)h (order)228 1419 y(obstructions.)d(Langston)14 b(et)g(al.)g(found)h (that)f(testing)f(for)i(just)f(one)h(of)f(these)h(ga)o(v)o(e)e(a)228 1473 y(new)22 b(heuristic)f(algorithm)f(sup)q(erior)h(to)g(those)h (then)g(curren)o(tly)f(in)g(use)h(in)g(VLSI)228 1527 y(applications)f([)p Fs(LR91)q Fx(].)42 b(Similar)21 b(ideas)i(ha)o(v)o(e)f(b)q(een)i(explored)f(b)o(y)g(Gustedt)f(and)228 1581 y(Steger)15 b([)p Fs(GS94)p Fx(].)303 1662 y Fs(4.1.)25 b(A)14 b(Useful)f(P)o(arameterized)g(Algorithm)g(for)h(MAST.)22 b Fx(W)l(e)12 b(next)h(de-)228 1716 y(scrib)q(e)j(a)f(useful)g(direct)h Fr(FPT)e Fx(algorithm)f(for)i(the)h Fm(Maximum)h(A)o(greement)h(Sub-) 228 1770 y(tree)k(\(MAST\))c Fx(problem)h(de\014ned)i(in)e Fl(x)g Fx(1.)32 b(Apart)18 b(from)h(the)g(in)o(trinsic)f(in)o(terest) 228 1824 y(of)d(this)g(result,)g(it)f(is)i(a)f(nice)h(example)f(of)g(t) o(w)o(o)f(imp)q(ortan)o(t)g(p)q(oin)o(ts)h(concerning)h Fr(FPT)228 1878 y Fx(algorithms.)315 1941 y Fl(\017)21 b Fx(Our)d(algorithm)d(for)j(MAST)f(ultimately)f(uses)i(an)g(algorithm) d(for)i Fm(Ver)m(tex)359 1995 y(Co)o(ver)f Fx(as)f(a)h(subroutine.)22 b(Useful)16 b Fr(FPT)f Fx(algorithms)f(lead)i(to)f(other)h(useful)359 2049 y Fr(FPT)e Fx(algorithms,)f(as)h(one)i(migh)o(t)e(naturally)g(exp) q(ect.)315 2103 y Fl(\017)21 b Fx(There)e(is)g(already)f(a)h(p)q (olynomial)f(time)g(algorithm)f(for)i(MAST)g(for)f(b)q(oun-)359 2157 y(ded)c(degree)g(trees,)f(so)h(wh)o(y)f(b)q(other)h(with)f(an)h (exp)q(onen)o(tial)f Fr(FPT)g Fx(algorithm?)359 2211 y(The)21 b(answ)o(er)g(is)g(that)g(the)g(p)q(olynomial)f(time)h (algorithm)e(for)i(MAST)g(due)359 2265 y(to)g([)p Fs(FPT95)p Fx(])h(runs)g(in)g(time)g Fj(O)q Fx(\()p Fj(r)q(n)1008 2249 y Ft(3)1042 2265 y Fx(+)15 b Fj(n)1119 2249 y Fh(d)1140 2265 y Fx(\))21 b(for)h Fj(r)h Fx(ph)o(ylogenetic)f(trees)g(of)359 2319 y(maxim)o(um)14 b(degree)i Fj(d)g Fx(on)g(a)f(set)h(of)f Fj(n)h Fx(sp)q(ecies.)22 b(The)16 b(algorithm)e(w)o(e)h(describ)q(e)359 2373 y(requires)f(time)h Fj(O)q Fx(\()p Fj(c)707 2357 y Fh(k)727 2373 y Fj(r)q(n)8 b Fx(log)f Fj(n)p Fx(\))15 b(where)h Fj(c)e Fx(is)h(a)g(constan)o(t)f(less)h(than)g(3.)k(Conse-) 359 2427 y(quen)o(tly)l(,)13 b(this)g(is)g(an)g(example)h(of)f(a)g (situation)f Fr(wher)n(e)j(a)g(classic)n(al)r(ly)e(exp)n(onen-)359 2481 y(tial)h(FPT)g(algorithm)h(may)g(b)n(e)f(pr)n(efer)n(able)g(to)h (a)g(p)n(olynomial)f(time)h(algorithm)p Fx(.)228 2554 y Fs(Theorem.)k Fx(The)14 b(parameterized)f(MAST)g(problem)h(can)g(b)q (e)g(solv)o(ed)g(in)f(time)g(b)q(oun-)228 2608 y(ded)j(b)o(y)f Fj(O)q Fx(\()p Fj(c)451 2591 y Fh(k)471 2608 y Fj(r)q(n)8 b Fx(log)f Fj(n)p Fx(\))15 b(for)g Fj(r)h Fx(trees)f(on)g Fj(n)g Fx(sp)q(ecies.)p eop %%Page: 15 15 15 14 bop 228 213 a Fs(Sk)o(etc)o(h.)21 b Fx(The)c(input)f(to)f(the)h (problem)g(is)g(a)f(set)h(of)g(ro)q(oted)g(binary)f(trees)h Fj(T)1579 220 y Ft(1)1598 213 y Fj(;)8 b(:::;)g(T)1706 220 y Fh(r)228 267 y Fx(eac)o(h)k(ha)o(ving)g Fj(n)h Fx(lea)o(v)o(es)f(lab)q(eled)g(in)h(1:1)e(corresp)q(ondence)j(with)e(a) g(set)g Fj(X)k Fx(of)c Fj(n)h Fx(sp)q(ecies.)228 321 y(The)g(problem)g(is)g(to)f(determine)i(if)e(it)h(is)f(p)q(ossible)h (to)g(delete)g(at)g(most)f Fj(k)i Fx(sp)q(ecies)g(from)228 374 y Fj(X)i Fx(to)d(obtain)g(a)g(set)g Fj(X)623 358 y Fi(0)647 374 y Fx(on)g(whic)o(h)g(all)g(of)f(the)i(trees)f Fr(agr)n(e)n(e)p Fx(.)19 b(In)14 b(considering)e(this)h(pro-)228 428 y(blem,)h(there)i(is)e(an)h(elegan)o(t)f(p)q(oin)o(t)h(of)f(view)h (dev)o(elop)q(ed)h(b)o(y)f(Bry)o(an)o(t)f([)p Fs(Bry97)o Fx(])h(based)228 482 y(on)i Fr(triples)p Fx(.)25 b(If)18 b Fl(f)p Fj(a;)8 b(b;)g(c)p Fl(g)14 b Fx(is)j(a)g(set)g(of)f(three)i (sp)q(ecies)f(in)h Fj(X)t Fx(,)e(then)i(the)f(restriction)f(of)228 536 y(eac)o(h)i(of)f(the)h(trees)g Fj(T)607 543 y Fh(i)639 536 y Fx(to)f(these)h(three)g(sp)q(ecies)h(m)o(ust)e(b)q(e)i(one)f(of)f (the)h(three)g(p)q(ossi-)228 590 y(ble)d(alternativ)o(es)e(\(using)i (paren)o(thetical)f(notation)g(to)h(represen)o(t)g(trees\):)k(\()p Fj(a;)8 b Fx(\()p Fj(b;)g(c)p Fx(\)\),)228 644 y(\()p Fj(b;)g Fx(\()p Fj(a;)g(c)p Fx(\))o(\),)16 b(\()p Fj(c;)8 b Fx(\()p Fj(a;)g(b)p Fx(\)\),)16 b(or)i(\()p Fj(a;)8 b(b;)g(c)p Fx(\).)28 b(If)19 b(t)o(w)o(o)e(or)h(more)h(of)f(these)h (three)g(p)q(ossibilities)228 698 y(arise)14 b(among)g(the)h Fj(T)585 705 y Fh(i)599 698 y Fx(,)f(then)i(ob)o(viously)d(it)i(will)e (b)q(e)j(necessary)f(to)f(eliminate)g(at)g(least)228 752 y(one)d(of)g(the)g(sp)q(ecies)g(in)g Fl(f)p Fj(a;)d(b;)g(c)p Fl(g)g Fx(from)i Fj(X)15 b Fx(in)c(order)f(to)h(obtain)f(an)h(agreemen) o(t)f(subtree.)228 806 y(In)16 b(this)e(situation)g(w)o(e)h(will)f (refer)h(to)f Fl(f)p Fj(a;)8 b(b;)g(c)p Fl(g)13 b Fx(as)h(a)h Fr(c)n(on\015icte)n(d)g(triple)g Fx(of)g(sp)q(ecies.)303 860 y(An)20 b(argumen)o(t)f(due)i(to)e(Bry)o(an)o(t)g([)p Fs(Bry98)o Fx(])h(sho)o(ws)f(that)g(our)h(problem)f(can)h(b)q(e)228 914 y(reduced)15 b(in)e(this)g(w)o(a)o(y)g(to)g(the)h(w)o(ell-kno)o(wn) e Fm(3-Hitting)j(Set)f Fx(problem)f(whic)o(h)h(tak)o(es)228 968 y(as)f(input)g(a)h(collection)e Fl(C)k Fx(of)d(3-elemen)o(t)g (subsets)h(of)f(a)g(set)g Fj(X)k Fx(and)c(a)h(p)q(ositiv)o(e)e(in)o (teger)228 1022 y Fj(k)q Fx(,)17 b(and)g(m)o(ust)g(answ)o(er)g(whether) g(there)g(is)g(a)f(subset)i Fj(X)1206 1006 y Fi(0)1233 1022 y Fl(\022)e Fj(X)k Fx(with)c Fl(j)p Fj(X)1502 1006 y Fi(0)1513 1022 y Fl(j)f(\024)i Fj(k)h Fx(suc)o(h)228 1076 y(that)12 b(for)g(eac)o(h)h Fj(A)g Fl(2)g(C)s Fx(,)f Fj(A)5 b Fl(\\)g Fj(X)749 1060 y Fi(0)774 1076 y Fl(6)p Fx(=)12 b Fl(;)p Fx(.)19 b(Bry)o(an)o(t's)12 b(argumen)o(t)g(sho)o(ws)g (that)g(our)g(problem)228 1130 y(is)j(equiv)m(alen)o(t)g(to)g (\014nding)h(a)f Fj(k)q Fx(-elemen)o(t)h(hitting)e(set)h(for)g(the)h (triples)f(of)g Fj(X)k Fx(that)14 b(are)228 1184 y(con\015icted)h(with) g(resp)q(ect)h(to)e(the)h Fj(T)852 1191 y Fh(i)866 1184 y Fx(.)303 1238 y(The)20 b(set)f(of)h(con\015icted)g(triples)f(could)h (b)q(e)h(computed)f(b)o(y)g(exhaustiv)o(ely)f(com-)228 1292 y(puting)c(the)h(restrictions)f(of)g(the)h Fj(T)846 1299 y Fh(i)876 1292 y Fx(for)f(eac)o(h)h(3-elemen)o(t)g(subset)g(of)f Fj(X)t Fx(,)g(but)h(this)g(is)228 1346 y(not)f(v)o(ery)h(e\016cien)o (t.)22 b(In)16 b(time)g Fj(O)q Fx(\()p Fj(n)8 b Fx(log)f Fj(n)p Fx(\))15 b(it)h(is)f(p)q(ossible)h(to)f(either)h(determine)g (that)228 1400 y(t)o(w)o(o)g(trees)i(are)f(isomorphic,)h(or)f(iden)o (tify)g(a)g(con\015icted)i(triple.)26 b(Once)19 b(a)f(con\015icted)228 1454 y(triple)12 b(is)g(iden)o(ti\014ed,)i(w)o(e)e(can)h(branc)o(h)h (in)e(a)h(searc)o(h)g(tree)g(based)g(on)g(3)g(p)q(ossibilities)e(for) 228 1508 y(resolving)g(the)h(con\015ict.)19 b(F)l(or)11 b(example,)h(if)g(the)g(con\015icted)g(triple)f(is)h Fl(f)p Fj(a;)c(b;)g(c)p Fl(g)h Fx(then)j(w)o(e)228 1562 y(create)g(three)g(branc)o(hes)h(in)f(the)h(searc)o(h)f(tree)g(b)o(y)h (deleting)f(one)g(elemen)o(t)g(\(sa)o(y)g Fj(a)p Fx(\))g(from)228 1616 y Fj(X)18 b Fx(and)d(from)g(all)f(of)g(the)i(trees)e Fj(T)808 1623 y Fh(i)822 1616 y Fx(.)20 b(W)l(e)15 b(no)o(w)g(recursiv) o(ely)f(attempt)g(to)h(determine)g(if)228 1670 y(the)g(mo)q(di\014ed)h (instance)f(can)g(b)q(e)h(solv)o(ed)f(with)f Fj(k)1079 1653 y Fi(0)1103 1670 y Fx(=)f Fj(k)f Fl(\000)e Fx(1.)303 1724 y(There)22 b(will)f(b)q(e)i(at)e(most)g Fj(O)q Fx(\(3)860 1707 y Fh(k)881 1724 y Fx(\))h(no)q(des)h(in)f(the)g(searc)o(h)g(tree,) h(and)f(the)h(run-)228 1778 y(ning)13 b(time)h(due)g(to)f(eac)o(h)h(no) q(de)h(is)e Fj(O)q Fx(\()p Fj(r)q(n)8 b Fx(log)f Fj(n)p Fx(\),)14 b(whic)o(h)f(yields)h(the)g(claimed)f(running)228 1832 y(time.)1358 b Fd(2)228 1939 y Fx(The)19 b(algorithm)d(sk)o(etc)o (hed)j(ab)q(o)o(v)o(e)f(uses)h Fm(3-Hitting)h(Set)f Fx(implicitly)e(as) h(a)g(means)228 1993 y(of)f(solving)g(the)h(parameterized)f(MAST)h (problem.)27 b(An)18 b(impro)o(v)o(ed)f(but)h(more)f(ela-)228 2047 y(b)q(orate)i Fr(FPT)f Fx(algorithm)f(for)i(this)f(problem)h(is)g (describ)q(ed)h(in)f([)p Fs(BFRS98)p Fx(],)h(where)228 2101 y(w)o(e)c(reduce)i(MAST)e(to)g Fm(3-Hitting)i(Set)p Fx(,)f(and)g(in)f(turn)h(devise)f(a)h(go)q(o)q(d)f(algorithm)228 2155 y(for)i Fm(3-Hitting)j(Set)e Fx(using)f(a)h(subroutine)g(for)f Fm(Ver)m(tex)k(Co)o(ver)p Fx(,)e(to)e(obtain)g(an)228 2210 y(algorithm)11 b(with)j Fj(c)e Fx(=)h(\(1)7 b(+)706 2172 y Fl(p)p 744 2172 46 2 v 38 x Fx(17)o(\))p Fj(=)p Fx(2.)19 b(A)13 b(heuristic)g(algorithm)f(that)h(migh)o(t)g(b)q(e)h (useful)228 2264 y(for)g(a)g(greater)g(range)h(of)f(the)h(parameter)f Fj(k)i Fx(can)f(b)q(e)g(adapted)g(from)f(the)h(ab)q(o)o(v)o(e)f Fr(FPT)228 2317 y Fx(algorithm)f(simply)h(b)o(y)i(not)e(exploring)h (all)f(the)h(branc)o(hes)h(of)e(the)i(searc)o(h)f(tree.)303 2446 y Fs(4.2.)25 b(The)19 b(Steiner)e(Problem)g(for)h(Generalized)g (Hyp)q(ercub)q(es.)k Fx(In)17 b(this)228 2500 y(section)22 b(w)o(e)g(brie\015y)h(describ)q(e)g(another)f Fr(FPT)f Fx(result)h(based)h(on)f(the)h(elemen)o(tary)228 2554 y(metho)q(d)c(of)g Fr(r)n(e)n(duction)h(to)g(a)h(pr)n(oblem)f(kernel)p Fx(.)31 b(The)20 b(basic)f(idea)g(of)g(this)f(metho)q(d,)228 2608 y(whic)o(h)g(is)g(v)o(ery)h(widely)f(applicable)g(\(see)g([)p Fs(DF98)p Fx(]\))g(is)g(that)g(b)o(y)g(v)m(arious)g(reduction)p eop %%Page: 16 16 16 15 bop 228 213 a Fx(rules)19 b(that)g(require)h(time)f Fr(p)n(olynomial)g(in)h(b)n(oth)h Fj(n)f Fr(and)h Fj(k)f Fx(it)f(ma)o(y)g(b)q(e)h(p)q(ossible)g(to)228 267 y(reduce)e(the)f (input)h(\()p Fj(x;)8 b(k)q Fx(\))16 b(for)g(a)h(parameterized)g (problem)g(to)g(an)g(instance)g(\()p Fj(x)1636 250 y Fi(0)1648 267 y Fj(;)8 b(k)1694 250 y Fi(0)1705 267 y Fx(\))228 321 y(where)21 b Fj(k)390 304 y Fi(0)423 321 y Fl(\024)g Fj(k)h Fx(and)f Fl(j)p Fj(x)658 304 y Fi(0)669 321 y Fl(j)f Fx(is)g(b)q(ounded)i(b)o(y)e(a)g(function)h(of)f Fj(k)q Fx(.)35 b(This)21 b(immediately)228 374 y(implies)15 b(that)g(the)h(problem)f(is)g(in)h Fr(FPT)p Fx(,)e(since)i(the)g (question)f(can)h(b)q(e)h(answ)o(ered)e(b)o(y)228 428 y(brute)h(force)f(for)g(\()p Fj(x)574 412 y Fi(0)585 428 y Fj(;)8 b(k)631 412 y Fi(0)642 428 y Fx(\).)21 b(Surprisingly)l(,) 15 b(it)g(can)h(b)q(e)g(sho)o(wn)f(that)g Fr(every)h Fx(problem)f(in)228 482 y Fr(FPT)f Fx(can)h(b)q(e)h(k)o(ernelized.)21 b(Suc)o(h)15 b(reductions)h(seem)f(to)g(b)q(e)h(closely)e(related)h(to) f(ideas)228 536 y(explored)j(in)g([)p Fs(CKT91,)i(KS94)p Fx(].)25 b(Since)18 b(suc)o(h)f(reductions)g(simplify)f(the)h(input,)g (it)228 590 y(seems)e(that)f(they)h(cannot)g(h)o(urt,)f(and)h(th)o(us)g (constitute)f(a)g(reasonable)h(\\prepro)q(cess-)228 644 y(ing")f(heuristic)h(regardless)f(of)h(what)g(comes)g(after)f(that.)303 698 y Fm(The)19 b(Steiner)i(Pr)o(oblem)g(f)o(or)f(Hyper)o(cubes)f Fx(is)f(of)f(in)o(terest)g(to)g(biologists)228 752 y(in)i(the)h (computation)f(of)g(ph)o(ylogenetic)g(trees)h(under)g(the)g(criterion)f (of)g(minim)o(um)228 806 y(ev)o(olution)13 b(/)h(maxim)o(um)f(parsimon) o(y)l(.)19 b(The)14 b(set)g Fj(S)j Fx(corresp)q(onds)d(to)g(a)g(set)g (of)g(sp)q(ecies,)228 860 y(and)19 b(the)g(binary)f(v)o(ectors)h (corresp)q(ond)g(to)f(information)f(ab)q(out)i(the)g(sp)q(ecies,)h(eac) o(h)228 914 y(comp)q(onen)o(t)11 b(recording)g(the)h(answ)o(er)e(to)h (some)g(question)g(\(as)g(0)g(or)f(1\),)h(suc)o(h)h(as:)18 b(\\Do)q(es)228 968 y(it)13 b(ha)o(v)o(e)h(wings?")20 b(or)14 b(\\Is)g(there)h(a)f(th)o(ymine)g(at)g(a)g(certain)f(p)q (osition)h(in)g(the)h(DNA)f(se-)228 1022 y(quence?")21 b(Eac)o(h)15 b(suc)o(h)g(bit)f(of)h(information)e(is)h(termed)h(a)f Fr(char)n(acter)i Fx(of)e(the)h(sp)q(ecies.)228 1076 y(In)h(realistic)d(applications)h(the)i(n)o(um)o(b)q(er)g Fj(k)g Fx(of)f(sp)q(ecies)h(ma)o(y)e(usefully)h(b)q(e)h(around)g(40)228 1130 y(or)g(60,)g(while)g(the)h(n)o(um)o(b)q(er)g(of)g(c)o(haracters)f Fj(n)h Fx(ma)o(y)f(b)q(e)h(v)o(ery)f(large.)24 b(W)l(e)17 b(consider)f(a)228 1184 y(sligh)o(tly)d(more)i(general)g(problem)f (than)i(the)f(one)g(describ)q(ed)h(in)f Fl(x)h Fx(1.)228 1292 y Fm(The)h(Steiner)g(Pr)o(oblem)i(f)o(or)e(Generalized)i(Hyper)o (cubes)228 1346 y Fk(Instance:)h Fx(The)14 b(input)h(to)f(the)g (problem)g(consists)g(of)g(the)g(follo)o(wing)e(pieces)j(of)f(infor-) 228 1400 y(mation:)228 1454 y(\(1\))i(A)h(set)g(of)f(complete)h(w)o (eigh)o(ted)f(digraphs)h Fj(D)1090 1461 y Fh(i)1120 1454 y Fx(for)g Fj(i)e Fx(=)h(1)p Fj(;)8 b(:::;)g(n)p Fx(,)14 b(eac)o(h)j(describ)q(ed)228 1508 y(b)o(y)e(a)g(set)g(of)g(v)o(ertices) f Fj(V)643 1515 y Fh(i)672 1508 y Fx(and)h(a)g(function)812 1581 y Fj(t)828 1588 y Fh(i)855 1581 y Fx(:)e Fj(V)908 1588 y Fh(i)931 1581 y Fl(\002)e Fj(V)1004 1588 y Fh(i)1030 1581 y Fl(!)i Fj(I)-12 b(N)228 1654 y Fx(\(W)l(e)18 b(refer)g(to)f(the) h(v)o(ertices)g(of)f Fj(D)834 1661 y Fh(i)866 1654 y Fx(as)h Fr(char)n(acter)i(states)p Fx(,)d(to)h Fj(D)1367 1661 y Fh(i)1399 1654 y Fx(as)f(the)i Fr(char)n(acter)228 1708 y(state)d(digr)n(aph)p Fx(,)f(and)h(to)e Fj(t)669 1715 y Fh(i)699 1708 y Fx(as)h(the)g Fr(state)h(tr)n(ansition)f(c)n (ost)h(function)f Fx(for)f(the)h Fj(i)p Fx(th)g(c)o(ha-)228 1762 y(racter.\))228 1815 y(\(2\))f(A)h(p)q(ositiv)o(e)f(in)o(teger)h Fj(k)693 1822 y Ft(1)727 1815 y Fx(suc)o(h)h(that)e Fl(j)p Fj(V)968 1822 y Fh(i)982 1815 y Fl(j)e(\024)h Fj(k)1079 1822 y Ft(1)1113 1815 y Fx(for)i Fj(i)d Fx(=)h(1)p Fj(;)8 b(:::;)g(n)p Fx(.)228 1869 y(\(3\))13 b(A)h(set)h Fj(X)i Fx(of)d Fj(k)549 1876 y Ft(2)583 1869 y Fx(length)g Fj(n)g Fx(v)o(ectors)g Fj(x)942 1876 y Fh(j)974 1869 y Fx(for)g Fj(j)h Fx(=)e(1)p Fj(;)8 b(:::;)g(k)1252 1876 y Ft(2)1269 1869 y Fx(,)14 b(where)h(the)f Fj(i)p Fx(th)g(comp)q(o-)228 1923 y(nen)o(t)h Fj(x)356 1930 y Fh(j)374 1923 y Fx([)p Fj(i)p Fx(])d Fl(2)h Fj(V)498 1930 y Fh(i)511 1923 y Fx(.)20 b(That)15 b(is,)f(for)h Fj(j)g Fx(=)e(1)p Fj(;)8 b(:::;)g(k)995 1930 y Ft(2)1012 1923 y Fx(,)830 2022 y Fj(x)856 2029 y Fh(j)886 2022 y Fl(2)13 b Fx(\012)g(=)1041 1979 y Fh(n)1040 2022 y Fj(=)-23 b Fl(n)1023 2063 y Fh(i)p Ft(=1)1080 2022 y Fj(V)1107 2029 y Fh(i)228 2124 y Fx(\(4\))14 b(A)h(p)q(ositiv)o(e)f(in)o(teger)h Fj(M)5 b Fx(.)228 2178 y Fk(P)o(arameter:)18 b Fx(\()p Fj(k)509 2185 y Ft(1)528 2178 y Fj(;)8 b(k)573 2185 y Ft(2)592 2178 y Fx(\))228 2232 y Fk(Question:)31 b Fx(Is)22 b(there)f(a)f(ro)q(oted)h (tree)g Fj(T)28 b Fx(=)23 b(\()p Fj(V)s(;)8 b(E)s Fx(\))18 b(and)j(an)g(assignmen)o(t)f(to)g(eac)o(h)228 2286 y(v)o(ertex)15 b Fj(v)f Fl(2)f Fj(V)25 b Fx(of)14 b Fj(T)21 b Fx(of)15 b(an)g(elemen)o(t)g Fj(y)901 2293 y Fh(v)935 2286 y Fl(2)d Fx(\012,)j(suc)o(h)h(that:)315 2350 y Fl(\017)21 b Fj(X)d Fx(is)d(assigned)g(1:1)f(with)g(the)i(set)f(of)f(lea)o(v)o(es)h(of)f Fj(T)6 b Fx(,)315 2404 y Fl(\017)21 b Fx(The)d(sum)g(o)o(v)o(er)g(all)f (paren)o(t-c)o(hild)g(edges)i Fj(uv)h Fx(of)d Fj(T)6 b Fx(,)19 b(of)f(the)g Fr(total)h(tr)n(ansition)359 2458 y(c)n(ost)14 b Fx(for)h(the)g(edge,)g(de\014ned)i(to)d(b)q(e)833 2508 y Fh(n)811 2522 y Fc(X)815 2620 y Fh(i)p Ft(=1)884 2565 y Fj(t)900 2572 y Fh(i)914 2565 y Fx(\()p Fj(y)954 2572 y Fh(u)977 2565 y Fx([)p Fj(i)p Fx(])p Fj(;)8 b(y)1062 2572 y Fh(v)1081 2565 y Fx([)p Fj(i)p Fx(]\))p eop %%Page: 17 17 17 16 bop 359 213 a Fx(is)14 b(b)q(ounded)j(b)o(y)e Fj(M)5 b Fx(?)228 303 y Fs(Theorem)14 b(1.)20 b Fm(The)14 b(Steiner)i(Pr)o (oblem)g(f)o(or)f(Generalized)h(Hyper)o(cubes)f Fx(is)228 357 y(\014xed-parameter)g(tractable.)228 465 y Fs(Pro)q(of.)22 b Fx(W)l(e)16 b(de\014ne)h(a)e(relation)g Fj(i)e Fl(\030)i Fj(j)j Fx(on)e(the)g(index)g(space)g Fl(f)p Fx(1)p Fj(;)8 b(:::;)g(n)p Fl(g)13 b Fx(that)i(allo)o(ws)228 519 y(us)20 b(to)f(com)o(bine)g Fj(D)571 526 y Fh(i)605 519 y Fx(and)h Fj(D)736 526 y Fh(j)773 519 y Fx(and)g(obtain)f(an)h(equiv)m(alen)o(t)g (smaller)e(instance.)33 b(T)l(o)228 573 y(de\014ne)16 b Fl(\030)g Fx(w)o(e)f(\014rst)f(de\014ne)j(some)e(other)f(relations.) 303 627 y(Fix)e Fj(m)g Fl(\024)h Fj(k)506 634 y Ft(1)539 627 y Fx(and)g(let)g Fj(l)g Fx(b)q(e)h(an)f(in)o(teger)f(edge)i(lab)q (eling)e(of)h(the)g(complete)g(digraph)228 681 y Fj(K)267 688 y Fh(m)319 681 y Fx(on)19 b Fj(m)h Fx(v)o(ertices.)31 b(Let)20 b Fj(v)748 688 y Ft(1)768 681 y Fj(;)8 b(:::;)g(v)871 688 y Fh(m)921 681 y Fx(denote)19 b(the)h(v)o(ertices)e(of)h Fj(K)1417 688 y Fh(m)1450 681 y Fx(.)33 b(Let)19 b Fj(T)25 b Fx(b)q(e)20 b(a)228 735 y(ro)q(oted)15 b(tree)g(with)f Fj(k)589 742 y Ft(2)623 735 y Fx(lea)o(v)o(es)h(lab)q(eled)g(from)f Fj(v)1038 742 y Ft(1)1058 735 y Fj(;)8 b(:::;)g(v)1161 742 y Fh(m)1192 735 y Fx(.)20 b(De\014ne)15 b(the)h Fr(c)n(ost)e Fx(of)h Fj(T)21 b Fx(with)228 788 y(resp)q(ect)16 b(to)e Fj(l)i Fx(to)f(b)q(e)h(the)f(minim)o(um,)g(o)o(v)o(er)f(all)g(p)q (ossible)i(lab)q(elings)e Fj(s)i Fx(of)f(the)g(in)o(ternal)228 842 y(v)o(ertices)d(of)h Fj(T)19 b Fx(with)13 b(lab)q(els)f(tak)o(en)h (from)g Fl(f)p Fj(v)986 849 y Ft(1)1005 842 y Fj(;)8 b(:::;)g(v)1108 849 y Fh(m)1139 842 y Fl(g)p Fx(,)13 b(of)f(the)i(sum)f(o)o(v)o(er)f(the)h(paren)o(t-)228 896 y(c)o(hild)k(edges)g(of)g Fj(T)23 b Fx(of)17 b(the)g(transition)f (costs)h(giv)o(en)f(b)o(y)h Fj(l)h Fx(on)f(the)h(lab)q(els,)f(and)g (write)228 950 y(this)d(as)675 1010 y(cost\()p Fj(T)t(;)8 b(l)q Fx(\))i(=)j(min)991 1017 y Fh(s)1009 1010 y Fl(f)p Fx(cost)o(\()p Fj(T)t(;)8 b(s;)g(l)q Fx(\))p Fl(g)228 1078 y Fx(If)17 b Fj(l)h Fx(and)f Fj(l)412 1062 y Fi(0)441 1078 y Fx(are)g(in)o(teger)f(edge)i(lab)q(elings)e(of)h Fj(K)1060 1085 y Fh(m)1110 1078 y Fx(and)h Fj(T)23 b Fx(is)17 b(as)f(ab)q(o)o(v)o(e,)h(then)h(de\014ne)228 1132 y Fj(l)13 b Fl(\030)290 1139 y Fh(T)330 1132 y Fj(l)345 1116 y Fi(0)372 1132 y Fx(if)h(and)i(only)e(if)h Fl(9)p Fj(s)h Fx(suc)o(h)f(that)452 1210 y(cost)o(\()p Fj(T)t(;)8 b(l)q Fx(\))i(=)j(cost\()p Fj(T)t(;)8 b(s;)g(l)q Fx(\))k(and)j(cost\()p Fj(T)t(;)8 b(s;)g(l)1221 1192 y Fi(0)1229 1210 y Fx(\))13 b(=)g(cost)o(\()p Fj(T)t(;)8 b(l)1471 1192 y Fi(0)1481 1210 y Fx(\))228 1288 y(De\014ne)16 b Fj(l)d Fl(\030)g Fj(l)459 1272 y Fi(0)485 1288 y Fx(if)i(and)g(only)g Fj(l)e Fl(\030)777 1295 y Fh(T)817 1288 y Fj(l)832 1272 y Fi(0)859 1288 y Fx(for)h(all)g(suc)o(h)i(trees)f Fj(T)6 b Fx(.)303 1342 y(F)l(or)18 b(a)h(\014xed)g(v)m(alue)g(of)g Fj(k)744 1349 y Ft(2)782 1342 y Fx(\(the)g(n)o(um)o(b)q(er)g(of)g(lea)o (v)o(es\))f Fj(l)h Fl(\030)g Fj(l)1361 1326 y Fi(0)1391 1342 y Fx(dep)q(ends)i(only)d(on)228 1396 y(\014nitely)h(man)o(y)f (trees)i Fj(T)25 b Fx(and)19 b(lab)q(elings)g Fj(s)p Fx(,)h(for)e(the)i(reason)f(that)f(if)h(a)g(tree)g Fj(T)25 b Fx(has)228 1450 y(a)e(\\v)o(ery)f(long")h(path)g(of)f(v)o(ertices)h (of)g(degree)g(2,)i(a)e(lab)q(eling)f Fj(s)h Fx(that)g(minimizes)228 1504 y(cost)o(\()p Fj(T)t(;)8 b(s;)g(l)q Fx(\))i(will)i(necessarily)h (b)q(e)h(constan)o(t)e(on)h(\\most)g(of)t(")f(the)h(in)o(ternal)f(v)o (ertices)h(of)228 1558 y(the)k(path.)24 b(In)18 b(more)e(detail,)g(sa)o (y)g(that)g(a)h(v)o(ertex)f(of)h Fj(T)22 b Fx(is)17 b Fr(imp)n(ortant)g Fx(if)f(it)g(is)g(either)228 1612 y(a)g(leaf)f(of)h Fj(T)22 b Fx(or)16 b(has)g(degree)g(more)g(than)g(2.)23 b(Note)16 b(that)f(if)h Fj(k)1293 1619 y Ft(2)1329 1612 y Fx(is)f(\014xed,)i(then)g(for)e(all)228 1666 y(of)h(the)h(trees)f Fj(T)23 b Fx(that)16 b(w)o(e)g(m)o(ust)g(consider)h(in)f(order)h(to)f (determine)h(whether)f Fj(l)g Fl(\030)f Fj(l)1698 1650 y Fi(0)1710 1666 y Fx(,)228 1720 y(the)j(n)o(um)o(b)q(er)h(of)e(imp)q (ortan)o(t)g(v)o(ertices)g(of)h Fj(T)24 b Fx(is)18 b(b)q(ounded)h(b)o (y)f(a)g(function)g(of)g Fj(k)1627 1727 y Ft(2)1646 1720 y Fx(.)29 b(A)228 1774 y(little)15 b(though)o(t)h(will)g(sho)o(w)g (that)g(w)o(e)h(need)g(only)g(consider)f Fj(T)23 b Fx(suc)o(h)17 b(that)f(if)h Fj(u)g Fx(and)g Fj(v)228 1828 y Fx(are)e(imp)q(ortan)o(t) f(v)o(ertices)h(that)g(are)g(joined)g(b)o(y)g(a)h(path)f(of)g(unimp)q (ortan)o(t)g(v)o(ertices)f(in)228 1882 y Fj(T)6 b Fx(,)14 b(then)i(the)f(length)g(of)g(the)g(path)g(is)g(at)g(most)f Fj(m)p Fx(.)303 1936 y(F)l(or)g Fj(i;)8 b(i)437 1920 y Fi(0)460 1936 y Fl(2)13 b(f)p Fx(1)p Fj(;)8 b(:::;)g(n)p Fl(g)k Fx(de\014ne)k Fj(i)c Fl(\030)h Fj(i)915 1920 y Fi(0)942 1936 y Fx(if)h(and)i(only)f(if:)303 2002 y(1.)20 b Fl(j)p Fj(V)399 2009 y Fh(i)412 2002 y Fl(j)14 b Fx(=)g Fl(j)p Fj(V)528 2010 y Fh(i)540 2000 y Fb(0)553 2002 y Fl(j)f Fx(=)i Fj(m)h Fx(so)g(that)f(the)h(only)g(di\013erence)g(b)q (et)o(w)o(een)h Fj(D)1440 2009 y Fh(i)1469 2002 y Fx(and)g Fj(D)1597 2010 y Fh(i)1609 2000 y Fb(0)1638 2002 y Fx(is)e(in)359 2056 y(their)f(arc-lab)q(elings)h Fj(l)g Fx(and)h Fj(l)865 2040 y Fi(0)876 2056 y Fx(,)e(and)303 2110 y(2.)20 b Fj(l)13 b Fl(\030)g Fj(l)449 2094 y Fi(0)475 2110 y Fx(.)228 2176 y(The)19 b(k)o(ernelization)e(algorithm)f(can)j(no)o(w)f(b)q(e)h (describ)q(ed)h(quite)e(simply)l(.)30 b(Let)18 b Fj(I)k Fx(b)q(e)228 2230 y(an)e(instance)g(of)g(the)h(problem.)35 b(If)21 b(there)f(are)h(indices)f Fj(i)h Fl(6)p Fx(=)h Fj(i)1347 2214 y Fi(0)1378 2230 y Fx(for)e(whic)o(h)h Fj(i)g Fl(\030)g Fj(i)1698 2214 y Fi(0)1710 2230 y Fx(,)228 2284 y(then)g(mo)q(dify)f Fj(I)k Fx(b)o(y)d(com)o(bining)f(these)h(in)o (to)f(one)h(c)o(haracter)f(state)f(digraph)i(with)228 2338 y(the)15 b(state)f(transition)g(cost)g(function)h(giv)o(en)g(b)o (y)g(the)g(arc-lab)q(eling)f(de\014ned)i(as)f Fj(l)10 b Fx(+)g Fj(l)1698 2321 y Fi(0)1710 2338 y Fx(,)228 2392 y(where)20 b(these)h(are)f(the)g(cost)g(functions)g(for)f Fj(D)1062 2399 y Fh(i)1096 2392 y Fx(and)i Fj(D)1228 2400 y Fh(i)1240 2390 y Fb(0)1253 2392 y Fx(,)g(resp)q(ectiv)o(ely)l(.) 34 b(Because)228 2446 y Fj(i)12 b Fl(\030)h Fj(i)320 2429 y Fi(0)346 2446 y Fx(there)i(is)f(no)g(p)q(ossible)h(harm)f(in)g 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y(go)q(o)q(d)j(and)h(can)g(probably)f(b)q(e)i(m)o(uc)o(h)f(impro)o (v)o(ed.)32 b(W)l(e)20 b(remark)f(that)g(most)g(of)g(the)228 482 y(exp)q(ense)f(is)f(in)g(determining)f(when)i(t)o(w)o(o)e (transition)f(digraph)i(indices)g Fj(i)g Fx(and)g Fj(i)1633 466 y Fi(0)1661 482 y Fx(are)228 536 y(equiv)m(alen)o(t)d(b)o(y)h (testing)f(them)g(on)h(`the)f(relev)m(an)o(t)h(set)f(of)g(trees)h(with) f Fj(k)1452 543 y Ft(2)1486 536 y Fx(lea)o(v)o(es.)19 b(This)228 590 y(suggests)f(a)h(heuristic)g(algorithm)e(that)h(com)o (bines)h(indices)g(when)h(they)f(fail)f(to)g(b)q(e)228 644 y(distinguished)11 b(b)o(y)h(a)f(\(m)o(uc)o(h)h(smaller\))e(random) h(sample)h(of)f(trees)h(and)g(leaf-lab)q(elings.)303 698 y(In)22 b Fl(x)h Fx(1)f(w)o(e)f(rep)q(orted)i(an)f(encoun)o(ter)g (with)g(an)g(ev)o(olutionary)e(biologist)g(who)228 752 y(rep)q(orted)c(earlier,)f(rather)h(fruitless)f(in)o(teractions)g(with) g(theoretical)g(computer)h(sci-)228 806 y(en)o(tists)f(who)i(pro)o(v)o (ed)f(that)f(his)i(problems)f(w)o(ere)g 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y(miniaturization)d(of)h(Co)q(ok's)h(Theorem.)303 2554 y Fm(Clique)j Fx(is)g(a)g(t)o(ypical)g(starting)f(p)q(oin)o(t)h (for)g Fj(W)6 b Fx([1]-hardness)18 b(argumen)o(ts.)29 b(It)19 b(is)228 2608 y(in)o(teresting)d(that)g(most)h(natural)f (parameterized)h(problems)g(seem)g(to)g(b)q(elong)g(to)f(a)p eop %%Page: 19 19 19 18 bop 228 173 1507 2 v 462 211 a Fm(Linear)17 b(Inequalities)280 264 y Fj(W)6 b Fx([)p Fj(P)g Fx(])72 b Fm(Minimum)16 b(Axiom)h(Set)462 318 y(Shor)m(t)g(Sa)m(tisfiability)462 372 y(Weighted)h(Cir)o(cuit)f(Sa)m(tisfiability)p 228 390 V 249 428 a Fj(W)6 b Fx([)p Fj(S)s(AT)g Fx(])40 b Fm(Weighted)18 b(Sa)m(tisfiability)p 228 446 V 462 484 a(Longest)f(Common)g(Subsequence)492 538 y(\()p Fj(k)d Fx(=)f Fl(h)k Fm(number)h(of)f(seqs.,)p Fl(j)p Fx(\006)p Fl(ji)p Fm(\))c Fx(\(hard\))283 592 y Fj(W)6 b Fx([)p Fj(t)p Fx(],)75 b Fm(Feasible)18 b(Register)h(Assignment)14 b Fx(\(hard\))260 646 y(for)h(all)f Fj(t)53 b Fm(Triangula)m(ting)17 b(Colored)h(Graphs)d Fx(\(hard\))462 700 y 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(b)o(y)h(pro)q(cesses)f(that)g(inhabit)228 1263 y(suc)o(h)k(zones)h (themselv)o(es)f(\(e.g.,)f(computer)h(co)q(de)h(that)e(is)h(written)f (in)h(a)g(structured)228 1317 y(manner)j(so)g(that)f(it)g(can)i(b)q(e)f (comprehensible)h(to)e(the)h(programmer\),)g(and)g(these)228 1371 y(distributions)c(then)i(inherit)f(limited)f(parameter)h(ranges)g (b)q(ecause)h(of)f(the)h(compu-)228 1424 y(tational)e(parameters)h (that)h(implicitly)e(go)o(v)o(ern)h(the)h(feasibilit)o(y)e(of)i(the)g (generativ)o(e)228 1478 y(pro)q(cesses,)c(though)f(the)g(relev)m(an)o (t)g(parameters)g(ma)o(y)f(not)h(b)q(e)h(immediately)e(ob)o(vious.)228 1586 y(It)j(seems)g(that)g(w)o(e)g(ha)o(v)o(e)g(a)f(whole)h(new)h(w)o (orld)e(of)h(complexit)o(y)f(issues)h(to)f(explore!)852 1759 y Fs(References)228 1832 y Fp([ADF95])19 b(K.)10 b(Abrahamson,)j(R.)d(Do)o(wney)i(and)g(M.)e(F)m(ello)o(ws,)g(\\Fixed)i (P)o(arameter)f(T)m(ractabilit)o(y)f(and)307 1877 y(Completeness)g(IV:) g(On)g(Completeness)g(for)g Fa(W)5 b Fp([)p Fa(P)g Fp(])10 b(and)h Fo(PSP)m(A)o(CE)f Fp(Analogs,")g Fo(A)o(nnals)f(of)i(Pur)n(e) 307 1923 y(and)h(Applie)n(d)f(L)n(o)n(gic)h Fp(73)h(\(1995\),)g (235{276.)228 1969 y([AF)o(GPR96])19 b(E.)c(Allender,)h(J.)f(F)m(eigen) o(baum,)h(J.)f(Goldsmith,)h(T.)f(Pitassi)h(and)g(S.)f(Rudic)o(h,)i (\\The)307 2014 y(F)m(uture)g(of)f(Computational)h(Complexit)o(y)g (Theory:)24 b(P)o(art)17 b(I)q(I,")f Fo(SIGA)o(CT)g(News)g Fp(27)h(\(1996\),)307 2060 y(3{7.)228 2106 y([AL97])i(E.)c(Aarts)h(and) g(J.)f(K.)g(Lenstra)i(\(eds.\),)f Fo(L)n(o)n(c)n(al)f(Se)n(ar)n(ch)f (in)i(Combinatorial)e(Optimization)p Fp(,)307 2151 y(John)f(Wiley)g (and)h(Sons,)f(1997.)228 2197 y([ALMSS92])19 b(S.)e(Arora,)g(C.)e (Lund,)j(R.)f(Mot)o(w)o(ani,)g(M.)f(Sudan)i(and)f(M.)f(Szegedy)m(,)i (\\Pro)q(of)f(V)m(eri\014-)307 2242 y(cation)h(and)h(In)o(tractabilit)o (y)e(of)h(Appro)o(ximation)h(Algorithms,")g Fo(Pr)n(o)n(c)n(e)n(e)n (dings)c(of)j(the)f(IEEE)307 2288 y(Symp)n(osium)12 b(on)h(the)f(F)m (oundations)e(of)j(Computer)g(Scienc)n(e)p Fp(,)d(13{22,)k(1992.)228 2334 y([A)m(YZ94])19 b(N.)e(Alon,)h(R.)g(Y)m(uster)g(and)h(U.)e(Zwic)o (k,)i(\\Color-Co)q(ding:)27 b(A)17 b(New)g(Metho)q(d)i(for)f(Find-)307 2379 y(ing)24 b(Simple)g(P)o(aths,)j(Cycles)d(and)h(Other)f(Small)g (Subgraphs)i(Within)f(Large)f(Graphs,")307 2425 y Fo(Pr)n(o)n(c.)13 b(Symp.)f(The)n(ory)h(of)g(Computing)f(\(STOC\))p Fp(,)h(A)o(CM)f (\(1994\),)i(326{335.)228 2471 y([Ba94])19 b(B.)j(Bak)o(er,)k(\\Appro)o (ximation)d(Algorithms)g(for)f Fo(NP)p Fp(-Complete)g(Problems)h(on)g (Planar)307 2516 y(Graphs,")14 b Fo(J.A.C.M.)e Fp(41)h(\(1994\),)g (153{180.)228 2562 y([Baz95])19 b(C.)d(Bazgan,)j(\\Sc)o(h)o(\023)-18 b(emas)17 b(d'appro)o(ximation)h(et)f(complexit)o(\023)-18 b(e)17 b(param)o(\023)-18 b(etr)o(\023)g(ee,")18 b(Rapp)q(ort)g(de)307 2608 y(stage)13 b(de)g(DEA)g(d'Informatique)h(\022)-19 b(a)13 b(Orsa)o(y)m(,)f(1995.)p eop %%Page: 23 23 23 22 bop 228 213 a Fp([BDFHW95])20 b(H.)9 b(Bo)q(dlaender,)i(R.)f(Do)o (wney)m(,)g(M.)g(F)m(ello)o(ws,)f(M.)g(Hallett)g(and)i(H.)e(T.)g(W)m (areham,)i(\\P)o(a-)307 258 y(rameterized)g(Complexit)o(y)g(Analysis)g (in)g(Computational)g(Biology)m(,")g Fo(Computer)g(Applic)n(ations)307 304 y(in)i(the)f(Bioscienc)n(es)e Fp(11)j(\(1995\),)g(49{57.)228 350 y([BFR98])19 b(R.)9 b(Balasubramanian,)j(M.)d(F)m(ello)o(ws)g(and)h (V.)f(Raman,)i(\\An)e(Impro)o(v)o(ed)h(Fixed-P)o(arameter)307 395 y(Algorithm)i(for)h(V)m(ertex)g(Co)o(v)o(er,")g Fo(Information)e (Pr)n(o)n(c)n(essing)g(L)n(etters)h Fp(65)h(\(1998\),)g(163{168.)228 441 y([BFRS98])20 b(D.)12 b(Bry)o(an)o(t,)h(M.)g(F)m(ello)o(ws,)f(V.)g (Raman)h(and)h(U.)e(Stege,)h(\\On)g(the)g(P)o(arameterized)g(Com-)307 487 y(plexit)o(y)g(of)f(MAST)h(and)h(3-Hitting)e(Sets,")h(man)o (uscript,)h(1998.)228 532 y([Bo)q(d96])20 b(H.)14 b(Bo)q(dlaender,)i (\\A)f(Linear)g(Time)f(Algorithm)h(for)f(Finding)i(T)m(ree)e(Decomp)q (ositions)i(of)307 578 y(Small)c(T)m(reewidth,")h Fo(SIAM)g(J.)g(Comp.) g Fp(25)g(\(1996\),)g(1305{1317.)228 624 y([Bry97])19 b(D.)11 b(Bry)o(an)o(t,)g(\\Building)h(T)m(rees,)e(Hun)o(ting)i(for)e (T)m(rees,)h(and)g(Comparing)h(T)m(rees,")e(Ph.D.)h(dis-)307 669 y(sertation,)31 b(Departmen)o(t)e(of)e(Mathematics,)32 b(Univ.)27 b(Can)o(terbury)m(,)32 b(Christc)o(h)o(urc)o(h,)g(New)307 715 y(Zealand,)13 b(1997.)228 761 y([Bry98])19 b(D.)13 b(Bry)o(an)o(t,)g(priv)n(ate)g(comm)o(unication,)h(1998.)228 806 y([CC97])k(L.)e(Cai)g(and)h(J.)e(Chen.)h(\\On)h(Fixed-P)o(arameter) g(T)m(ractabilit)o(y)e(and)i(Appro)o(ximabilit)o(y)f(of)307 852 y Fo(NP)p Fp(-Hard)i(Optimization)g(Problems,")j Fo(J.)d(Computer)g(and)g(Systems)f(Scienc)n(es)f Fp(54)j(\(1997\),)307 898 y(465{474.)228 943 y([CCDF96])g(L.)8 b(Cai,)h(J.)f(Chen,)i(R.)f(G.) f(Do)o(wney)i(and)g(M.)e(R.)h(F)m(ello)o(ws,)g(\\On)g(the)g(P)o (arameterized)g(Com-)307 989 y(plexit)o(y)15 b(of)f(Short)i (Computation)g(and)f(F)m(actorization,")h Fo(A)o(r)n(ch.)e(for)h(Math.) g(L)n(o)n(gic)f Fp(36)h(\(1997\),)307 1035 y(321{337.)228 1080 y([CCDF97])k(L.)10 b(Cai,)h(J.)g(Chen,)h(R.)f(Do)o(wney)h(and)g (M.)f(F)m(ello)o(ws,)f(\\Advice)i(Classes)f(of)g(P)o(arameterized)307 1126 y(T)m(ractabilit)o(y)m(,")h Fo(A)o(nnals)f(of)i(Pur)n(e)g(and)g (Applie)n(d)e(L)n(o)n(gic)h Fp(84)h(\(1997\),)g(119{138.)228 1172 y([Ces96])19 b(M.)9 b(Cesati,)h(\\Structural)h(Asp)q(ects)f(of)g (P)o(arameterized)g(Complexit)o(y)m(,")g(Ph.D.)g(dissertation,)307 1217 y(Univ)o(ersit)o(y)j(of)f(Rome,)h(1995.)228 1263 y([CKT91])18 b(P)m(.)13 b(Cheeseman,)h(B.)f(Kanefsky)h(and)g(W.)f(T)m (a)o(ylor,)g(\\Where)h(the)g(Really)f(Hard)h(Problems)307 1308 y(Are,")9 b Fo(Pr)n(o)n(c.)h(12th)f(Internationa)o(l)f(Joint)h (Confer)n(enc)n(e)f(on)i(A)o(rti\014cial)e(Intel)r(ligenc)m(e)e Fp(\(1991\),)k(331{)307 1354 y(337.)228 1400 y([CT97])18 b(M.)10 b(Cesati)g(and)i(L.)e(T)m(revisan,)g(\\On)h(the)g(E\016ciency)g (of)f(P)o(olynomial)h(Time)e(Appro)o(ximation)307 1445 y(Sc)o(hemes,")k Fo(Information)f(Pr)n(o)n(c)n(essing)f(L)n(etters)g Fp(64)j(\(1997\),)f(165{171.)228 1491 y([DKL96])19 b(T.)9 b(Dean,)i(J.)e(Kirman)g(and)i(S.-H.)d(Lin,)i(\\Theory)h(and)f(Practice) g(in)f(Planning,")i(T)m(ec)o(hnical)307 1537 y(Rep)q(ort,)i(Computer)h (Science)f(Departmen)o(t,)h(Bro)o(wn)f(Univ)o(ersit)o(y)m(,)f(1996.)228 1582 y([DEF93])19 b(R.)e(Do)o(wney)m(,)h(P)m(.)f(Ev)n(ans)h(and)f(M.)g (F)m(ello)o(ws,)g(\\P)o(arameterized)h(Learning)f(Complexit)o(y)m(,") 307 1628 y Fo(Pr)n(o)n(c.)c(6th)f(A)o(CM)i(Workshop)e(on)h (Computational)e(L)n(e)n(arning)h(The)n(ory)g Fp(\(1993\),)h(51{57.)228 1674 y([DF95])19 b(R.)14 b(G.)f(Do)o(wney)h(and)h(M.)e(R.)g(F)m(ello)o (ws,)g(\\Fixed)h(P)o(arameter)g(T)m(ractabilit)o(y)f(and)h(Complete-) 307 1719 y(ness)f(I:)g(Basic)g(Theory)m(,")g Fo(SIAM)g(Journal)f(of)h (Computing)e Fp(24)i(\(1995\),)g(873{921.)228 1765 y([DF98])19 b(R.)f(G.)f(Do)o(wney)i(and)g(M.)e(R.)h(F)m(ello)o(ws,)g Fo(Par)n(ameterize)n(d)e(Complexity)p Fp(,)h(Springer-V)m(erlag,)307 1811 y(1998.)228 1856 y([DFKHW94])i(R.)9 b(G.)g(Do)o(wney)m(,)h(M.)f(F) m(ello)o(ws,)g(B.)g(Kapron,)h(M.)f(Hallett,)g(and)h(H.)f(T.)f(W)m (areham.)i(\\The)307 1902 y(P)o(arameterized)k(Complexit)o(y)f(of)g (Some)h(Problems)g(in)f(Logic)h(and)g(Linguistics,")f Fo(Pr)n(o)n(c)n(e)n(e)n(dings)307 1948 y(Symp)n(osium)i(on)i(L)n(o)n (gic)n(al)e(F)m(oundations)f(of)i(Computer)h(Scienc)n(e)d(\(LF)o(CS\))p Fp(,)i(Springer-V)m(erlag,)307 1993 y(Lecture)d(Notes)g(in)g(Computer)g (Science)h(v)o(ol.)e(813)i(\(1994\),)f(89{100.)228 2039 y([DFS99])19 b(R.)f(G.)g(Do)o(wney)m(,)h(M.)f(R.)g(F)m(ello)o(ws,)g (and)g(U.)g(Stege.)g(\\P)o(arameterized)h(Complexit)o(y:)26 b(A)307 2085 y(F)m(ramew)o(ork)16 b(for)f(Systematically)h(Confron)o (ting)g(P)o(arameterized)g(Complexit)o(y)m(,")g(in)g Fo(The)g(F)m(u-)307 2130 y(tur)n(e)c(of)g(Discr)n(ete)g(Mathematics:)i (Pr)n(o)n(c)n(e)n(e)n(dings)c(of)j(the)f(First)g(DIMA)m(TIA)h(Symp)n (osium,)e(Stirin)307 2176 y(Castle,)h(Cze)n(ch)h(R)n(epublic,)f(June,)g (1997)p Fp(,)f(F.)i(Rob)q(erts,)h(J.)e(Krato)q(c)o(h)o(vil)h(and)h(J.)e (Ne)n(\024)-17 b(set)n(\024)g(ril)12 b(\(eds.\),)307 2222 y(AMS-DIMA)o(CS)h(Pro)q(ceedings)h(Series,)f(1999)g(\()p Fo(in)g(print)p Fp(\).)228 2267 y([F)m(el97])18 b(J.)13 b(F)m(elsenstein.)f(Priv)n(ate)h(comm)o(unication,)h(1997.)228 2313 y([FK93])19 b(M.)e(F)m(ello)o(ws)g(and)h(N.)f(Koblitz,)h (\\Fixed-P)o(arameter)h(Complexit)o(y)e(and)i(Cryptograph)o(y)m(,")307 2359 y Fo(Pr)n(o)n(c)n(e)n(e)n(dings)c(of)i(the)h(10th)e(Intl.)h(Symp.) g(on)g(Applie)n(d)f(A)o(lgebr)n(a,)h(A)o(lgebr)n(aic)e(A)o(lgorithms)h (and)307 2404 y(Err)n(or-Corr)n(e)n(cting)11 b(Co)n(des)p Fp(,)g(Springer-V)m(erlag,)i(Berlin,)f(Lecture)g(Notes)h(in)f(Computer) h(Science)307 2450 y(v)o(ol.)f(673)i(\(1993\),)f(121{131.)228 2496 y([FPT95])19 b(M.)c(F)m(arac)o(h,)i(T.)e(Przyt)o(yc)o(k)n(a,)j (and)e(M.)g(Thorup.)g(\\On)h(the)f(agreemen)o(t)g(of)g(man)o(y)h (trees")307 2541 y Fo(Information)11 b(Pr)n(o)n(c)n(essing)g(L)n (etters)i(55)f Fp(\(1995\),)h(297{301.)p eop %%Page: 24 24 24 23 bop 228 213 a Fp([F)m(r97])19 b(J.)12 b(F)m(ranco,)h(J.)f (Goldsmith,)h(J.)f(Sc)o(hlipf,)g(E.)h(Sp)q(ec)o(k)o(enmey)o(er)h(and)g (R.P)m(.)e(Sw)o(aminathan,)i(\\An)307 258 y(Algorithm)f(for)h(the)f (Class)h(of)f(Pure)h(Implicational)g(F)m(orm)o(ulas,")g(to)f(app)q(ear) i(in)f Fo(Discr)n(ete)e(Ap-)307 304 y(plie)n(d)g(Mathematics)p Fp(.)228 350 y([GJ79])19 b(M.)13 b(Garey)h(and)g(D.)g(Johnson.)g Fo(Computers)g(and)f(Intr)n(actabilit)o(y:)j(A)d(Guide)g(to)h(the)f (The)n(ory)307 395 y(of)g Fp(NP)p Fo(-c)n(ompletenes)o(s)p Fp(.)d(W.)i(H.)h(F)m(reeman,)g(San)g(F)m(rancisco,)g(1979.)228 441 y([GS94])19 b(J.)11 b(Gustedt)h(and)g(A.)f(Steger,)h(\\T)m(esting)f (Hereditary)h(Prop)q(erties)g(E\016cien)o(tly)f(on)h(Av)o(erage,")307 487 y Fo(Or)n(ders,)21 b(A)o(lgorithms)d(and)h(Applic)n(atio)o(ns:)27 b(Pr)n(o)n(c)n(e)n(e)n(dings)17 b(of)j(the)f(Internation)o(al)e (Workshop)307 532 y(ORD)o(AL'94)p Fp(,)f(Springer-V)m(erlag,)i(Lecture) g(Notes)f(in)g(Computer)g(Science,)i(v)o(ol.)d(831)i(\(1994\),)307 578 y(100{116.)228 624 y([HGS98])h(M.)12 b(Hallett,)g(G.)h(Gonnet,)g (and)g(U.)f(Stege,)h(\\V)m(ertex)g(Co)o(v)o(er)g(Revisited:)k(A)c (Hybrid)g(Algo-)307 669 y(rithm)f(of)h(Theory)h(and)f(Heuristic,")g (man)o(uscript)g(1998.)228 715 y([HL92])19 b(J.)13 b(Hartmanis)i(and)g (H.)e(Lin,)i(editors,)f Fo(Computing)g(the)g(F)m(utur)n(e:)19 b(A)14 b(Br)n(o)n(ader)g(A)n(genda)f(for)307 761 y(Computer)g(Scienc)n (e)e(and)h(Engine)n(ering)p Fp(,)e(National)j(Academ)o(y)g(Press,)g (1992.)228 806 y([IW97])19 b(R.)12 b(Impagliazzo)g(and)g(A.)f (Wigderson,)i Fo(Pr)n(o)n(c.)f(A)o(CM)h(Symp)n(osium)e(on)g(The)n(ory)h (of)g(Comput-)307 852 y(ing)p Fp(,)f(1997.)228 898 y([JMc97])19 b(D.)14 b(S.)g(Johnson)h(and)g(L.)e(A.)h(McGeo)q(c)o(h,)h(\\The)f(T)m (ra)o(v)o(eling)g(Salesman)h(Problem:)k(A)14 b(Case)307 943 y(Study)m(,")j(in:)22 b(Aarts)16 b(and)g(Lenstra)h(\(eds.\),)f Fo(L)n(o)n(c)n(al)f(Se)n(ar)n(ch)f(in)i(Combinatorial)e(Optimization)p Fp(,)307 989 y(John)f(Wiley)g(and)h(Sons,)f(1997.)228 1035 y([JPY88])18 b(D.)f(S.)g(Johnson,)h(C.)e(H.)h(P)o(apadimitriou)g (and)h(M.)e(Y)m(annak)n(akis,)i(\\Ho)o(w)f(Easy)g(is)g(Lo)q(cal)307 1080 y(Searc)o(h?")h Fo(Journal)12 b(of)h(Computer)g(and)g(System)f (Scienc)n(es)e Fp(37)j(\(1988\),)g(79{100.)228 1126 y([Karp86])19 b(R.)14 b(M.)h(Karp,)g(\\Com)o(binatorics,)g(Complexit)o(y)f(and)h (Randomness,")i Fo(Communic)n(ations)307 1172 y(of)c(the)g(A)o(CM)g Fp(29)g(\(1986\),)g(98{109.)228 1217 y([Kr90])19 b(M.)e(W.)h(Kren)o (tel,)h(\\On)g(Finding)f(and)h(V)m(erifying)e(Lo)q(cally)h(Optimal)g (Solutions,")i Fo(SIAM)307 1263 y(Journal)12 b(on)h(Computing)e Fp(19)i(\(1990\),)g(742{751.)228 1308 y([KS94])19 b(S.)14 b(Kirkpatric)o(k)g(and)h(B.)e(Selman,)i(\\Critical)e(Beha)o(vior)h(in)g (the)g(Satis\014abilit)o(y)g(of)g(Bo)q(olean)307 1354 y(F)m(orm)o(ulae,")f Fo(Scienc)n(e)d Fp(264)j(\(1994\),)g(1297{1301.) 228 1400 y([Lev86])19 b(L.)d(Levin,)h(\\Av)o(erage)g(Case)g(Complete)f (Problems,")i Fo(SIAM)e(J.)h(Computing)d Fp(15)j(\(1986\),)307 1445 y(285{286.)228 1491 y([LR91])i(M.)j(Langston)h(and)g(S.)f(Ramac)o (handram)o(urthi,)k(\\Dense)e(La)o(y)o(outs)f(for)f(Series-P)o(arallel) 307 1537 y(Graphs,")14 b Fo(Pr)n(o)n(c.)f(Gr)n(e)n(at)f(L)n(akes)h (Symp)n(osium)e(on)i(VLSI)f Fp(\(1991\),)h(14{17.)228 1582 y([MR95a])19 b(R.)12 b(Mot)o(w)o(ani)h(and)g(P)m(.)e(Ragha)o(v)n (an,)j Fo(R)n(andomize)n(d)d(A)o(lgorithms)p Fp(,)f(Cam)o(bridge)i (Univ.)g(Press,)307 1628 y(1995.)228 1674 y([MR95b])20 b(S.)12 b(Maha)r(jan)h(and)g(H.)f(Ramesh,)h(\\Derandomizing)h (Semide\014nite)f(Programming)g(Based)307 1719 y(on)f(Appro)o(ximation) h(Algorithms,")g(in)f Fo(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)i(the)h(36th)e (A)o(nnual)g(IEEE)i(Symp)n(osium)307 1765 y(on)g(F)m(oundation)o(s)e (of)i(Computer)g(Scienc)n(e)d Fp(\(1995\),)j(162{169.)228 1811 y([MR98])19 b(M.)11 b(Maha)r(jan)i(and)e(V.)g(Raman,)h(\\P)o (arameterizing)g(Ab)q(o)o(v)o(e)g(the)f(Guaran)o(tee:)17 b(MaxSat)12 b(and)307 1856 y(MaxCut,")h(to)g(app)q(ear)h(in)f Fo(J.)g(A)o(lgorithms)p Fp(.)228 1902 y([NR99])19 b(R.)d(Niedermeider)g (and)i(P)m(.)e(Rossmanith,)i(\\Upp)q(er)f(Bounds)h(for)e(V)m(ertex)g (Co)o(v)o(er)h(F)m(urther)307 1948 y(Impro)o(v)o(ed",)e Fo(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)i(the)g(16th)g(Symp)n(osium)f(on)h (The)n(or)n(etic)n(al)e(Asp)n(e)n(cts)h(in)h(Computer)307 1993 y(Scienc)n(e)d(\(ST)m(A)o(CS'99\))p Fp(,)f(LNCS,)j(1999.)228 2039 y([PGWRS96])20 b(C.)12 b(P)o(apadimitriou,)g(O.)g(Goldreic)o(h,)g (A.)g(Wigderson,)h(A.)e(Razb)q(oro)o(v)j(and)f(M.)f(Sipser,)307 2085 y(\\The)d(F)m(uture)g(of)f(Computational)i(Complexit)o(y)e (Theory:)16 b(P)o(art)8 b(I,")h Fo(SIGA)o(CT)g(News)f Fp(27)h(\(1996\),)307 2130 y(6{12.)228 2176 y([PY96])19 b(C.)8 b(P)o(apadimitriou)h(and)h(M.)e(Y)m(annak)n(akis,)j(\\On)e (Limited)f(Nondeterminism)i(and)f(the)g(Com-)307 2222 y(plexit)o(y)h(of)f(the)i(V)o(C)e(Dimension,")i Fo(J.)g(Computer)f(and) g(Systems)f(Scienc)n(es)e Fp(53)j(\(1996\),)h(161{170.)228 2267 y([Rag88])19 b(P)m(.)13 b(Ragha)o(v)n(an,)i(\\Probabilistic)f (Construction)g(of)f(Deterministic)h(Algorithms:)k(Appro)o(x-)307 2313 y(imating)e(P)o(ac)o(king)g(In)o(teger)g(Programs,")i Fo(J.)e(Computer)g(and)f(Systems)g(Scienc)n(es)e Fp(37)j(\(1988\),)307 2359 y(130{143.)228 2404 y([RS85])j(N.)8 b(Rob)q(ertson)j(and)e(P)m(.)g (D.)f(Seymour,)j(\\Graph)f(Minors:)15 b(A)8 b(Surv)o(ey)m(,")j(in)e Fo(Surveys)e(in)j(Combi-)307 2450 y(natorics)p Fp(,)g(I.)i(Anderson,)h (ed.)g(\(Cam)o(bridge)g(Univ)o(ersit)o(y)f(Press:)17 b(Cam)o(bridge,)c(1985\),)g(153{171.)228 2496 y([Ste99])19 b(U.)f(Stege,)j(\\)e(Resolving)h(Con\015icts)f(in)g(Problems)g(in)g (Computational)h(Bio)q(c)o(hemistry",)307 2541 y(man)o(uscript)13 b(\(dissertation)h(draft\),)e(1999.)p eop %%Page: 25 25 25 24 bop 228 213 a Fp([SF99])19 b(U.)11 b(Stege)h(and)g(M.)f(F)m(ello) o(ws,)g(\\An)g(Impro)o(v)o(ed)h(Fixed-P)o(arameter-T)m(ractable)g (Algorithm)f(for)307 258 y(V)m(ertex)h(Co)o(v)o(er",)h(T)m(ec)o(h.)g (Rep.)g(318,)g(ETH)f(Zuric)o(h,)i(1999.)228 304 y([Th97])19 b(M.)f(Thorup,)h(\\Structured)h(Programs)e(Ha)o(v)o(e)g(Small)g(T)m (ree-Width)g(and)h(Go)q(o)q(d)g(Register)307 350 y(Allo)q(cation,")13 b Fo(Pr)n(o)n(c)n(e)n(e)n(dings)e(23r)n(d)j(Internation)o(al)d (Workshop)i(on)g(Gr)n(aph-The)n(or)n(etic)e(Conc)n(epts)307 395 y(in)k(Computer)g(Scienc)n(e,)e(WG'97)p Fp(,)h(R.)h(M\177)-19 b(ohring)16 b(\(ed.\),)g(Springer)g(V)m(erlag,)f(Lecture)g(Notes)g(in) 307 441 y(Computer)e(Science)h(v)o(ol.)e(1335)i(\(1997\),)f(318{332.) 228 487 y([W)m(ang97a])20 b(J.)15 b(W)m(ang.)i(Av)o(erage-case)g(in)o (tractable)f(NP)g(problems.)h(In)f(D.-Z.)g(Du)h(and)g(K.-I.)d(Ko,)307 532 y(editors,)d Fo(A)n(dvanc)n(es)f(in)h(L)n(anguages,)f(A)o (lgorithms,)g(and)h(Complexity,)g(Kluwer)h(A)n(c)n(ademic)e(Pub-)307 578 y(lishers,)h Fp(pp.)i(313-378,)g(1997.)228 624 y([W)m(ang97b])20 b(J.)10 b(W)m(ang.)i(Av)o(erage-case)g(computational)f(complexit)o(y)h (theory)m(.)f(In)g(L.)f(Hemaspaandra)307 669 y(and)16 b(A.)e(Selmen,)i(editors,)g Fo(Complexity)e(The)n(ory)h(R)n(etr)n(osp)n (e)n(ctive)d(II,)j Fp(Springer,)h(pp.)g(295-328,)307 715 y(1997.)228 761 y([Y)m(an95])j(M.)f(Y)m(annak)n(akis.)h(\\P)o(ersp) q(ectiv)o(es)h(on)f(Database)h(Theory)m(,")g Fo(Pr)n(o)n(c)n(e)n(e)n (dings)c(of)i(the)g(IEEE)307 806 y(Symp)n(osium)12 b(on)h(the)f(F)m (oundations)e(of)j(Computer)g(Scienc)n(e)e Fp(\(1995\),)i(224{246.)228 852 y([YV70])18 b(A.)13 b(S.)g(Y)m(essenin-V)m(olpin,)f(\\The)i (Ultrain)o(tuitionistic)e(Criticism)f(and)j(the)f(An)o(titraditional) 307 898 y(Program)g(for)g(F)m(oundations)h(of)f(Mathematics,")g(in:)k Fo(Intuitionism)11 b(and)h(Pr)n(o)n(of)h(The)n(ory:)k(Pr)n(o-)307 943 y(c)n(e)n(e)n(dings)c(of)h(the)h(Summer)g(Confer)n(enc)n(e)e(at)i (Bu\013alo,)f(N.Y.,)g(1968)p Fp(,)g(A.)g(Kino,)h(J.)f(Myhill)g(and)307 989 y(R.)e(E.)h(V)m(esley)g(\(eds.\),)f(North-Holland,)h(1970.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF