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y(applies)34 b(to)g(sub)s(exp)s(onen)m(tial)g(complexit)m(y)-8 b(.)47 b(The)35 b(connection)g(will)d(b)s(e)i(made)g(explicit)257 4666 y(in)e(the)h(last)f(section)h(of)f(the)h(article.)404 4786 y(The)42 b(article)e(is)h(organized)g(as)h(follo)m(ws:)60 b(After)42 b(in)m(tro)s(ducing)e(our)i(notation,)g(w)m(e)257 4907 y(start)48 b(with)f(a)g(brief)g(in)m(tro)s(duction)f(in)m(to)h (parameterized)g(complexit)m(y)g(theory)-8 b(.)89 b(In)257 5027 y(Section)35 b FA(??)p FB(,)g(w)m(e)g(in)m(tro)s(duce)g(the)g (M-hierarc)m(h)m(y)g(and)f(establish)g(the)h(connections)g(b)s(e-)257 5147 y(t)m(w)m(een)29 b(the)f(M-hierarc)m(h)m(y)g(and)f(sub)s(exp)s (onen)m(tial)g(time)f(complexit)m(y)g(on)h(the)g(one)h(hand,)257 5268 y(and)42 b(b)s(et)m(w)m(een)h(the)e(M-hierarc)m(h)m(y)h(and)f(the) h(W-hierarc)m(h)m(y)g(on)f(the)g(other)g(hand.)70 b(In)257 5388 y(Section)43 b FA(??)o FB(,)i(w)m(e)f(study)f(the)g(miniaturized)d (problems)h(that)h(originally)d(led)j(to)g(the)p eop %%Page: 4 4 4 3 bop 257 573 a FB(in)m(tro)s(duction)37 b(of)g(the)h(class)g(M)p Fx([1])p FB(.)59 b(W)-8 b(e)39 b(pro)m(v)m(e)f(a)g(n)m(um)m(b)s(er)g (of)f(completeness)i(results)257 693 y(for)27 b(M)q Fx([1])p FB(,)h(whic)m(h)h(are)e(based)i(on)e(a)h(com)m(binatorial)c(lemma)h (kno)m(wn)k(as)f(the)g Fz(Sp)-5 b(arsi\034c)g(a-)257 814 y(tion)33 b(L)-5 b(emma)36 b FB([)p FA(?)p FB(].)43 b(\(The)31 b(pro)s(of)e(of)h(the)h(Sparsi\034cation)d(Lemma)h(itself)g (is)h(b)s(ey)m(ond)h(the)257 934 y(scop)s(e)36 b(of)f(this)g (article.\))50 b(W)-8 b(e)36 b(put)f(these)i(results)e(in)g(the)g (wider)h(con)m(text)g(of)f(the)h(syn-)257 1054 y(tactically)26 b(de\034ned)k(complexit)m(y)d(class)i(SNP)f(in)g(Section)g FA(??)p FB(.)42 b(Finally)-8 b(,)26 b(in)h(Section)h FA(??)p FB(,)257 1175 y(w)m(e)k(translate)f(the)g(results)h(bac)m(k)g (to)e(the)i(w)m(orld)e(of)h(classical)e(complexit)m(y)i(theory)g(and) 257 1295 y(the)i(exp)s(onen)m(tial)f(time)f(h)m(yp)s(othesis.)404 1422 y(One)38 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y(stead)d(of)f(\020strongly)g(uniform)f (\034xed-parameter)h(tractabilit)m(y\021.)52 b(W)-8 b(e)37 b(prefer)g(to)f(w)m(ork)257 4592 y(with)k(strongly)f(uniform)g (\034xed-parameter)h(tractabilit)m(y)d(as)k(it)e(has)h(a)g(more)f (robust)257 4713 y(theory)-8 b(.)257 4960 y FA(2.1)8 b(.)46 b(Prop)s(ositional)35 b(Logic)257 5147 y FB(F)-8 b(orm)m(ulas)31 b(of)h Fz(pr)-5 b(op)g(ositional)34 b(lo)-5 b(gic)37 b FB(are)c(built)e(up)i(from)e Fz(pr)-5 b(op)g(ositional)33 b(variables)40 b Fy(X)3429 5162 y Fu(1)3468 5147 y FB(,)257 5268 y Fy(X)338 5283 y Fu(2)378 5268 y Fy(;)17 b(:)g(:)g(:)31 b FB(b)m(y)h(taking)f(conjunctions,)h(disjunctions,)g(and)g(negations.) 42 b(The)33 b(negation)e(of)257 5388 y(a)36 b(form)m(ula)e Fy(\013)j FB(is)e(denoted)i(b)m(y)g Fw(:)p Fy(\013)q FB(.)54 b(W)-8 b(e)36 b(distinguish)f(b)s(et)m(w)m(een)j Fz(smal)5 b(l)37 b(c)-5 b(onjunctions)p FB(,)p eop %%Page: 5 5 5 4 bop 257 573 a FB(denoted)32 b(b)m(y)g Fw(^)p FB(,)g(whic)m(h)g(are) 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b(of)f(all)f(Bo)s(olean)g(circuits)h(b)m(y)j(CIR)m (C.)f(Usually)-8 b(,)27 b(w)m(e)i(do)f(not)f(distinguish)f(b)s(et)m(w)m (een)257 2357 y(form)m(ulas)31 b(and)i(circuits,)f(that)g(is,)h(w)m(e)g (view)h(PR)m(OP)f(as)g(a)f(sub)s(class)i(of)39 b(CIR)m(C.)404 2510 y(F)-8 b(or)33 b Fy(t)d Fw(\025)g Fx(0)p FB(,)k Fy(d)c Fw(\025)g Fx(1)k FB(w)m(e)h(inductiv)m(ely)e(de\034ne)i(the)f (follo)m(wing)d(classes)k Fx(\000)3018 2525 y Fv(t;d)3137 2510 y FB(and)f Fx(\001)3409 2525 y Fv(t;d)257 2631 y FB(of)e(prop)s(ositional)e(form)m(ulas:)1349 2594 y Fr(1)669 2949 y Fx(\000)730 2964 y Fu(0)p Fv(;d)853 2949 y Fx(=)d Fw(f)p Fy(\025)1063 2964 y Fu(1)1124 2949 y Fw(^)c Fy(:)17 b(:)g(:)22 b Fw(^)g Fy(\025)1495 2964 y Fv(c)1557 2949 y Fw(j)28 b Fy(c)f Fw(\024)i Fy(d;)k(\025)1956 2964 y Fu(1)1995 2949 y Fy(;)17 b(:)g(:)g(:)f(;)h(\025)2271 2964 y Fv(c)2338 2949 y FB(literals)m Fw(g)p Fy(;)648 3095 y Fx(\001)729 3110 y Fu(0)p Fv(;d)853 3095 y Fx(=)27 b Fw(f)p Fy(\025)1063 3110 y Fu(1)1124 3095 y Fw(_)c Fy(:)17 b(:)g(:)22 b Fw(_)g Fy(\025)1495 3110 y Fv(c)1557 3095 y Fw(j)28 b Fy(c)f Fw(\024)i Fy(d;)k(\025)1956 3110 y Fu(1)1995 3095 y Fy(;)17 b(:)g(:)g(:)f(;)h(\025)2271 3110 y Fv(c)2338 3095 y FB(literals)m Fw(g)p Fy(;)588 3268 y Fx(\000)649 3283 y Fv(t)p Fu(+1)p Fv(;d)853 3268 y Fx(=)27 b Fw(f)1006 3174 y Fp(^)1008 3385 y Fv(i)p Fo(2)p Fv(I)1133 3268 y Fy(\016)1176 3283 y Fv(i)1232 3268 y Fw(j)h Fy(I)40 b FB(\034nite)32 b(index)h(set)g(and)g Fy(\016)2268 3283 y Fv(i)2324 3268 y Fw(2)28 b Fx(\001)2499 3283 y Fv(t;d)2617 3268 y FB(for)k(all)f Fy(i)d Fw(2)g Fy(I)8 b Fw(g)p Fy(;)568 3537 y Fx(\001)649 3552 y Fv(t)p Fu(+1)p Fv(;d)853 3537 y Fx(=)27 b Fw(f)1006 3443 y Fp(_)1008 3654 y Fv(i)p Fo(2)p Fv(I)1133 3537 y Fy(\015)1184 3552 y Fv(i)1240 3537 y Fw(j)g Fy(I)40 b FB(\034nite)33 b(index)f(set)i(and) e Fy(\015)2283 3552 y Fv(i)2339 3537 y Fw(2)c Fx(\000)2494 3552 y Fv(t;d)2612 3537 y FB(for)k(all)e Fy(i)e Fw(2)g Fy(I)8 b Fw(g)p Fy(:)257 3943 y Fx(\000)318 3958 y Fu(2)p Fv(;)p Fu(1)449 3943 y FB(is)37 b(the)g(class)f(of)h(all)d(form)m(ulas) i(in)g Fz(c)-5 b(onjunctive)37 b(normal)h(form)p FB(,)f(whic)m(h)g(w)m (e)h(often)257 4064 y(denote)f(b)m(y)f(CNF.)g(F)-8 b(or)35 b Fy(d)d Fw(\025)i Fx(1)p FB(,)i Fx(\000)1528 4079 y Fu(1)p Fv(;d)1659 4064 y FB(is)f(the)h(class)g(of)f(all)f(form)m(ulas)g (in)h Fy(d)p Fz(-c)-5 b(onjunctive)257 4184 y(normal)34 b(form)p FB(,)e(whic)m(h)h(w)m(e)h(denote)f(b)m(y)h Fy(d)p FB(-CNF)o(.)404 4337 y(The)43 b Fz(size)50 b Fw(j)p Fy(\015)5 b Fw(j)42 b FB(of)h(a)f(circuit)g Fy(\015)48 b FB(is)42 b(the)h(n)m(um)m(b)s(er)h(of)e(no)s(des)h(plus)g(the)g(n)m(um)m(b)s(er) h(of)257 4458 y(edges;)34 b(th)m(us)g(for)e(form)m(ulas)g(the)h(size)g (is)f Fy(O)s Fx(\()p FB(n)m(um)m(b)s(er)g(of)g(no)s(des)q Fx(\))p FB(.)44 b(W)-8 b(e)33 b(usually)f(use)i(the)257 4578 y(letter)e Fy(m)g FB(to)g(denote)h(the)f(size)g(of)g(a)g(form)m (ula)e(or)i(circuit)e(and)j(the)f(letter)g Fy(n)g FB(to)f(denote)257 4699 y(the)i(n)m(um)m(b)s(er)g(of)f(v)-5 b(ariables.)p 257 4898 1296 4 v 370 4959 a Fn(1)407 4990 y FE(W)e(e)35 b(prefer)e(to)i(use)f Fm(\000)g FE(and)h Fm(\001)f FE(instead)h(of)f (the)h(more)f(common)f Fm(\005)i FE(and)g Fm(\006)f FE(to)g(denote)h (classes)257 5089 y(of)30 b(prop)r(ositional)d(form)n(ulas)h(\()p Fm(\000)i FE(for)e(c)p 1437 5102 37 4 v(onjunctions,)i Fm(\001)f FE(for)g(d)p 2172 5102 47 4 v(isjunctions\).)42 b(The)30 b(reason)d(is)i(that)h(w)n(e)257 5189 y(w)n(an)n(t)25 b(to)h(reserv)n(e)d Fm(\005)j FE(and)g Fm(\006)f FE(for)g(classes)g(of) g(form)n(ulas)f(of)i(predicate)f(logic.)35 b(Often)26 b(in)g(parameterized)257 5289 y(complexit)n(y)-7 b(,)35 b(it)f(is)g(necessary)d(to)j(jump)g(bac)n(k)f(and)g(forth)h(b)r(et)n(w) n(een)f(prop)r(ositional)g(and)g(predicate)257 5388 y(logic,)27 b(and)g(it)h(is)g(helpful)g(to)g(k)n(eep)f(them)h(strictly)f(separated) f(on)i(the)g(notational)e(lev)n(el.)p eop %%Page: 6 6 6 5 bop 257 573 a FC(3)8 b(.)53 b(F)-11 b(undamen)l(tals)43 b(of)h(P)l(arameterized)g(Complexit)l(y)h(Theory)257 820 y FA(3.1)8 b(.)46 b(P)m(arameterized)36 b(Problems)f(and)k(Fixed-P) m(arameter)e(T)-9 b(ractabilit)m(y)257 1006 y FB(As)49 b(it)f(is)g(common)f(in)g(complexit)m(y)h(theory)-8 b(,)53 b(w)m(e)c(describ)s(e)g(decision)f(problems)g(as)257 1127 y(languages)35 b(o)m(v)m(er)i(\034nite)e(alphab)s(ets)g Fx(\006)p FB(.)53 b(T)-8 b(o)36 b(distinguish)e(them)h(from)g (parameterized)257 1247 y(problems,)d(w)m(e)i(refer)f(to)f(problems)g Fy(Q)c Fw(\022)g Fx(\006)1888 1211 y Fo(\003)1960 1247 y FB(as)33 b Fz(classic)-5 b(al)33 b(pr)-5 b(oblems)p FB(.)404 1373 y(A)22 b Fz(p)-5 b(ar)g(ameterization)29 b FB(of)22 b Fx(\006)1410 1337 y Fo(\003)1473 1373 y FB(is)g(a)h(mapping)e Fy(\024)28 b Fx(:)f(\006)2232 1337 y Fo(\003)2300 1373 y Fw(!)g Fs(N)38 b FB(that)22 b(is)h(p)s(olynomial) c(time)257 1494 y(computable.)43 b(A)32 b Fz(p)-5 b(ar)g(ameterize)g(d) 34 b(pr)-5 b(oblem)39 b FB(\(o)m(v)m(er)33 b Fx(\006)p FB(\))g(is)f(a)g(pair)g Fx(\()p Fy(Q;)17 b(\024)p Fx(\))32 b FB(consisting)g(of)257 1614 y(a)37 b(set)h Fy(Q)e Fw(\022)g Fx(\006)796 1578 y Fo(\003)873 1614 y FB(and)h(a)g(parameterization)d Fy(\024)k FB(of)e Fx(\006)2189 1578 y Fo(\003)2229 1614 y FB(.)57 b(If)37 b Fx(\()p Fy(Q;)17 b(\024)p Fx(\))37 b FB(is)g(a)g(parameterized)257 1735 y(problem)26 b(o)m(v)m(er)i(the)g (alphab)s(et)e Fx(\006)p FB(,)i(then)g(w)m(e)g(call)d(strings)i Fy(x)h Fw(2)g Fx(\006)2605 1698 y Fo(\003)2672 1735 y Fz(instanc)-5 b(es)34 b FB(of)27 b Fy(Q)g FB(or)g(of)257 1855 y Fx(\()p Fy(Q;)17 b(\024)p Fx(\))29 b FB(and)g(the)h(n)m(um)m(b)s (ers)g Fy(\024)p Fx(\()p Fy(x)p Fx(\))f FB(the)g(corresp)s(onding)g Fz(p)-5 b(ar)g(ameters)p FB(.)42 b(Sligh)m(tly)27 b(abusing)257 1975 y(notation,)37 b(w)m(e)g(call)e(a)h(parameterized)g(problem)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))37 b FB(a)f Fz(p)-5 b(ar)g (ameterization)43 b FB(of)36 b(the)257 2096 y(classical)31 b(problem)h Fy(Q)p FB(.)404 2222 y(Usually)-8 b(,)35 b(when)h(represen)m(ting)g(a)f(parameterized)f(problem)g(w)m(e)i(do)f (not)g(men)m(tion)257 2343 y(the)40 b(underlying)e(alphab)s(et)h (explicitly)e(and)i(use)h(a)f(notation)e(as)j(illustrated)d(b)m(y)j (the)257 2463 y(follo)m(wing)30 b(examples.)257 2734 y FA(Example)37 b(3.1.)49 b FB(Recall)30 b(that)i(a)g Fz(vertex)j(c)-5 b(over)42 b FB(in)32 b(a)g(graph)g Fy(G)27 b Fx(=)h(\()p Fy(V)5 b(;)17 b(E)6 b Fx(\))32 b FB(is)g(a)g(subset)257 2854 y Fy(S)50 b Fw(\022)44 b Fy(V)63 b FB(suc)m(h)43 b(that)f(for)f(eac)m(h)i(edge)f Fw(f)p Fy(u;)17 b(v)t Fw(g)42 b(2)i Fy(E)6 b FB(,)44 b(either)e Fy(u)h Fw(2)h Fy(S)k FB(or)41 b Fy(v)47 b Fw(2)d Fy(S)6 b FB(.)71 b(The)257 2974 y Fz(p)-5 b(ar)g(ameterize)g(d)34 b(vertex)h(c)-5 b(over)34 b(pr)-5 b(oblem)39 b FB(is)32 b(de\034ned)i(as)e(follo)m(ws:) p 527 3150 2700 4 v 527 3710 4 561 v 577 3266 a Fz(p-)p Ft(Ver)-8 b(tex-Co)m(ver)864 3386 y Fz(Instanc)j(e:)104 b FB(A)32 b(graph)h Fy(G)f FB(and)h(a)f(natural)g(n)m(um)m(b)s(er)g Fy(k)f Fw(2)d Fs(N)9 b FB(.)773 3507 y Fz(Par)-5 b(ameter:)105 b Fy(k)s FB(.)872 3627 y Fz(Pr)-5 b(oblem:)104 b FB(Decide)33 b(if)f Fy(G)g FB(has)h(a)f(v)m(ertex)j(co)m(v)m(er)e(of)g(size)f Fy(k)s FB(.)p 3223 3710 V 527 3713 2700 4 v 257 3929 a FA(Example)37 b(3.2.)49 b FB(The)33 b Fz(p)-5 b(ar)g(ameterize)g(d)33 b(satis\034ability)g(pr)-5 b(oblem)33 b(for)h(Bo)-5 b(ole)g(an)33 b(cir)-5 b(cuits)257 4049 y FB(is)32 b(de\034ned)i(as)f(follo)m(ws:)p 645 4205 2464 4 v 645 4772 4 567 v 695 4327 a Fz(p-)p Ft(Sa)-8 b(t\()p FB(CIR)m(C)p Ft(\))982 4447 y Fz(Instanc)j(e:)104 b FB(A)33 b(Bo)s(olean)e(circuit)g Fy(\015)5 b FB(.)891 4568 y Fz(Par)-5 b(ameter:)105 b Fw(j)p Fx(v)-5 b(ar)o(\()p Fy(\015)5 b Fx(\))p Fw(j)p FB(.)990 4688 y Fz(Pr)-5 b(oblem:)104 b FB(Decide)33 b(if)f Fy(\015)37 b FB(is)32 b(satis\034able.)p 3105 4772 V 645 4775 2464 4 v 257 5002 a(More)i(generally)-8 b(,)33 b(for)g(a)g(class)h Fx(\000)f FB(of)g(circuits)g(or)g(form)m (ulas,)g(w)m(e)h(let)f Fz(p-)p Ft(Sa)-8 b(t)n Fx(\(\000\))33 b FB(denote)257 5122 y(the)g(restriction)f(of)g Fz(p-)p Ft(Sa)-8 b(t)n Fx(\()p FB(CIR)m(C)q Fx(\))33 b FB(to)f(instances)h Fy(\015)g Fw(2)28 b Fx(\000)p FB(.)404 5243 y Fz(p-)o Ft(Sa)-8 b(t)o Fx(\(\000\))39 b FB(is)f(a)h(parameterization)e(of)i (the)g(classical)f(problem)f Ft(Sa)-8 b(t)o Fx(\(\000\))p FB(.)63 b(There)257 5363 y(are)26 b(other)g(in)m(teresting)g (parameterizations)e(of)h Ft(Sa)-8 b(t)o Fx(\(\000\))p FB(,)27 b(and)f(w)m(e)h(will)c(see)k(some)f(later.)p eop %%Page: 7 7 7 6 bop 257 573 a FA(Example)37 b(3.3.)49 b FB(The)42 b Fz(weight)51 b FB(of)41 b(an)g(assignmen)m(t)g Fw(V)50 b FB(is)41 b(the)h(n)m(um)m(b)s(er)g(of)f(v)-5 b(ariables)257 693 y(set)37 b(to)f Ft(tr)m(ue)g FB(b)m(y)h Fw(V)8 b FB(.)55 b(A)36 b(circuit)g Fy(\015)41 b FB(is)36 b Fy(k)s Fz(-satis\034able)p FB(,)g(for)g(some)g Fy(k)h Fw(2)d Fs(N)9 b FB(,)44 b(if)35 b(there)i(is)e(a)257 814 y(satisfying)28 b(assignmen)m(t)f Fw(V)37 b FB(of)28 b(w)m(eigh)m(t)g Fy(k)j FB(for)d Fy(\015)5 b FB(.)42 b(The)29 b Fz(weighte)-5 b(d)30 b(satis\034ability)g(pr)-5 b(oblem)257 934 y Ft(WSa)d(t)o Fx(\(\000\))46 b FB(for)f(a)h(class)h Fx(\000)f FB(of)g(circuits)f (asks)j(whether)f(a)f(giv)m(en)h(circuit)e Fy(\015)56 b Fw(2)51 b Fx(\000)c FB(is)257 1054 y Fy(k)s FB(-satis\034able)32 b(for)g(a)g(giv)m(en)h Fy(k)s FB(.)43 b(W)-8 b(e)33 b(consider)g(the)g (follo)m(wing)d(parameterization:)p 645 1186 2464 4 v 645 1753 4 567 v 695 1308 a Fz(p-)p Ft(WSa)-8 b(t)n Fx(\(\000\))982 1428 y Fz(Instanc)j(e:)104 b Fy(\015)33 b Fw(2)28 b Fx(\000)k FB(and)h Fy(k)e Fw(2)d Fs(N)9 b FB(.)891 1549 y Fz(Par)-5 b(ameter:)105 b Fy(k)s FB(.)990 1669 y Fz(Pr)-5 b(oblem:)104 b FB(Decide)33 b(if)f Fy(\015)37 b FB(is)32 b Fy(k)s FB(-satis\034able.)p 3105 1753 V 645 1756 2464 4 v 257 1934 a FA(De\034nition)k(3.4.)49 b FB(Let)31 b Fx(\006)g FB(b)s(e)g(a)g(\034nite)f(alphab)s(et)g(and)h Fy(\024)c Fx(:)h(\006)2533 1898 y Fo(\003)2601 1934 y Fw(!)f Fs(N)46 b FB(a)30 b(parameteriza-)257 2054 y(tion.)299 2215 y(\(1\))48 b(An)34 b(algorithm)d Fs(A)61 b FB(with)34 b(input)f(alphab)s(et)g Fx(\006)i FB(is)e(an)h Fz(fpt-algorithm)h(with)h(r)-5 b(esp)g(e)g(ct)472 2336 y(to)34 b Fy(\024)e FB(if)f(there)h(is)f(a)h (computable)e(function)h Fy(f)39 b Fx(:)28 b Fs(N)42 b Fw(!)27 b Fs(N)47 b FB(suc)m(h)33 b(that)f(the)g(running)472 2456 y(time)f(of)h Fs(A)61 b FB(on)32 b(input)g Fy(x)h FB(is)1635 2670 y Fy(f)1694 2589 y Fp(\000)1739 2670 y Fy(\024)p Fx(\()p Fy(x)p Fx(\))1926 2589 y Fp(\001)1995 2670 y Fw(\001)21 b(j)p Fy(x)p Fw(j)2155 2629 y Fv(O)r Fu(\(1\))2305 2670 y Fy(:)299 2903 y FB(\(2\))48 b(A)36 b(parameterized)g(problem)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))37 b FB(is)f Fz(\034xe)-5 b(d-p)g(ar)g(ameter)37 b(tr)-5 b(actable)43 b FB(if)36 b(there)h(is)472 3024 y(an)32 b(fpt-algorithm)d(with)j(resp)s(ect)i(to)e Fy(\024)h FB(that)f(decides)i Fy(Q)p FB(.)472 3164 y(FPT)f(denotes)h(the)f(class) f(of)g(all)f(\034xed-parameter)h(tractable)g(problems.)3227 3128 y Fr(2)257 3385 y FA(Example)37 b(3.5.)49 b Fz(p-)o Ft(Sa)-8 b(t)o Fx(\()p FB(CIR)m(C)q Fx(\))33 b FB(is)f (\034xed-parameter)g(tractable.)404 3505 y(Indeed,)j(the)f(ob)m(vious)g (brute-force)g(searc)m(h)h(algorithm)c(decides)j(if)f(a)g(circuit)g Fy(\015)39 b FB(of)257 3626 y(size)33 b Fy(m)g FB(with)f Fy(n)h FB(v)-5 b(ariables)31 b(is)h(satis\034able)g(in)g(time)f Fy(O)s Fx(\(2)2327 3590 y Fv(n)2395 3626 y Fw(\001)22 b Fy(m)p Fx(\))p FB(.)404 3847 y(W)-8 b(e)38 b(lea)m(v)m(e)h(it)e(to)g (the)i(reader)f(to)g(sho)m(w)h(that)e Fz(p-)p Ft(Ver)-8 b(tex-Co)m(ver)37 b FB(is)h(also)f(\034xed-)257 3967 y(parameter)42 b(tractable.)71 b(On)42 b(the)g(other)g(hand,)j Fz(p-)p Ft(WSa)-8 b(t)n Fx(\(2)p FB(-CNF)p Fx(\))42 b FB(do)s(es)g(not)g(seem)257 4088 y(to)36 b(b)s(e)g(\034xed-parameter)f (tractable.)52 b(W)-8 b(e)36 b(shall)f(no)m(w)h(in)m(tro)s(duce)g(the)g (theory)g(to)f(giv)m(e)257 4208 y(evidence)f(for)e(this)g(and)h(other)g (in)m(tractabilit)m(y)d(results.)257 4449 y FA(3.2)8 b(.)46 b(Reductions)257 4629 y(De\034nition)36 b(3.6.)49 b FB(Let)31 b Fx(\()p Fy(Q;)17 b(\024)p Fx(\))30 b FB(and)g Fx(\()p Fy(Q)1761 4593 y Fo(0)1785 4629 y Fy(;)17 b(\024)1885 4593 y Fo(0)1908 4629 y Fx(\))30 b FB(b)s(e)g(parameterized)g(problems) f(o)m(v)m(er)j(the)257 4750 y(alphab)s(ets)26 b Fx(\006)g FB(and)g Fx(\006)1042 4714 y Fo(0)1065 4750 y FB(,)h(resp)s(ectiv)m (ely)-8 b(.)42 b(An)26 b Fz(fpt-r)-5 b(e)g(duction)33 b FB(\(more)25 b(precisely)-8 b(,)27 b Fz(fpt)i(many-)257 4870 y(one)h(r)-5 b(e)g(duction)7 b FB(\))28 b(from)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))28 b FB(to)f Fx(\()p Fy(Q)1636 4834 y Fo(0)1660 4870 y Fy(;)17 b(\024)1760 4834 y Fo(0)1783 4870 y Fx(\))28 b FB(is)f(a)h(mapping)e Fy(R)j Fx(:)e(\006)2642 4834 y Fo(\003)2710 4870 y Fw(!)g Fx(\(\006)2945 4834 y Fo(0)2969 4870 y Fx(\))3007 4834 y Fo(\003)3074 4870 y FB(suc)m(h)i(that:)299 5031 y(\(1\))48 b(F)-8 b(or)32 b(all)e Fy(x)e Fw(2)g Fx(\006)1029 4995 y Fo(\003)1102 5031 y FB(w)m(e)33 b(ha)m(v)m(e)h Fy(x)28 b Fw(2)g Fy(Q)56 b Fw(\()-17 b(\))55 b Fy(R)q Fx(\()p Fy(x)p Fx(\))28 b Fw(2)g Fy(Q)2423 4995 y Fo(0)2447 5031 y Fy(:)p 257 5098 1296 4 v 370 5159 a Fn(2)407 5189 y FE(The)36 b(notion)g(of)g(\034xed-parameter)e(tractabilit)n(y)h(w)n(e)g (in)n(tro)r(duce)h(here)f(is)h(kno)n(wn)g(as)f(\020strongly)257 5289 y(uniform)i(\034xed-parameter)e(tractabilit)n(y)-7 b(.\021)70 b(The)36 b(alternativ)n(e)g(notion)g(\020uniform)g (\034xed-parameter)257 5388 y(tractabilit)n(y\021)e(do)r(es)27 b(not)g(require)g(the)h(function)g Fl(f)36 b FE(to)28 b(b)r(e)g(computable.)p eop %%Page: 8 8 8 7 bop 299 573 a FB(\(2\))48 b Fy(R)34 b FB(is)e(computable)f(b)m(y)j (an)e(fpt-algorithm)d(with)j(resp)s(ect)i(to)e Fy(\024)p FB(.)299 726 y(\(3\))48 b(There)28 b(is)e(a)g(computable)f(function)h Fy(g)31 b Fx(:)d Fs(N)42 b Fw(!)28 b Fs(N)41 b FB(suc)m(h)28 b(that)e Fy(\024)2743 690 y Fo(0)2767 726 y Fx(\()p Fy(R)q Fx(\()p Fy(x)p Fx(\)\))h Fw(\024)i Fy(g)t Fx(\()p Fy(\024)p Fx(\()p Fy(x)p Fx(\)\))472 846 y FB(for)j(all)e Fy(x)f Fw(2)f Fx(\006)1004 810 y Fo(\003)1044 846 y Fy(:)404 1038 y FB(W)-8 b(e)40 b(write)g Fx(\()p Fy(Q;)17 b(\024)p Fx(\))41 b Fw(\024)1207 1002 y Fr(fpt)1340 1038 y Fx(\()p Fy(Q)1455 1002 y Fo(0)1478 1038 y Fy(;)17 b(\024)1578 1002 y Fo(0)1602 1038 y Fx(\))40 b FB(if)f(there)h(is)g(an)g (fpt-reduction)g(from)e Fx(\()p Fy(Q;)17 b(\024)p Fx(\))41 b FB(to)257 1159 y Fx(\()p Fy(Q)372 1123 y Fo(0)396 1159 y Fy(;)17 b(\024)496 1123 y Fo(0)519 1159 y Fx(\))p FB(,)24 b(and)e(w)m(e)g(write)g Fx(\()p Fy(Q;)17 b(\024)p Fx(\))27 b Fw(\021)1515 1123 y Fr(fpt)1636 1159 y Fx(\()p Fy(Q)1751 1123 y Fo(0)1774 1159 y Fy(;)17 b(\024)1874 1123 y Fo(0)1898 1159 y Fx(\))k FB(if)g Fx(\()p Fy(Q;)c(\024)p Fx(\))27 b Fw(\024)2393 1123 y Fr(fpt)2514 1159 y Fx(\()p Fy(Q)2629 1123 y Fo(0)2652 1159 y Fy(;)17 b(\024)2752 1123 y Fo(0)2776 1159 y Fx(\))k FB(and)h Fx(\()p Fy(Q)3129 1123 y Fo(0)3152 1159 y Fy(;)17 b(\024)3252 1123 y Fo(0)3276 1159 y Fx(\))27 b Fw(\024)3418 1123 y Fr(fpt)257 1296 y Fx(\()p Fy(Q;)17 b(\024)p Fx(\))p FB(.)41 b(W)-8 b(e)23 b(let)868 1215 y Fp(\002)909 1296 y Fx(\()p Fy(Q;)17 b(\024)p Fx(\))1162 1215 y Fp(\003)1204 1238 y Fr(fpt)1320 1296 y FB(b)s(e)23 b(the)h(class)f(of)g(parameterized)g(problems)f(fpt-reducible)257 1417 y(to)33 b Fx(\()p Fy(Q;)17 b(\024)p Fx(\))p FB(,)32 b(that)g(is,)942 1541 y Fp(\002)983 1622 y Fx(\()p Fy(Q;)17 b(\024)p Fx(\))1236 1541 y Fp(\003)1278 1564 y Fr(fpt)1398 1622 y Fx(=)1501 1541 y Fp(\010)1560 1622 y Fx(\()p Fy(Q)1675 1581 y Fo(0)1698 1622 y Fy(;)g(\024)1798 1581 y Fo(0)1821 1622 y Fx(\))1887 1537 y Fp(\014)1887 1597 y(\014)1948 1622 y Fx(\()p Fy(Q)2063 1581 y Fo(0)2086 1622 y Fy(;)g(\024)2186 1581 y Fo(0)2209 1622 y Fx(\))28 b Fw(\024)2352 1581 y Fr(fpt)2472 1622 y Fx(\()p Fy(Q;)17 b(\024)p Fx(\))2725 1541 y Fp(\011)2784 1622 y Fy(:)257 1811 y FB(F)-8 b(or)38 b(ev)m(ery)h(class)f(C)h(of)e(parameterized)h(problems,)g(w)m(e)i (de\034ne)f(C-hardness)g(and)f(C-)257 1931 y(completeness)c(of)e(a)g (parameterized)g(problem)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))33 b FB(in)e(the)i(usual)g(w)m(a)m(y)-8 b(.)257 2123 y FA(Example)37 b(3.7.)49 b FB(Recall)38 b(that)i(an)h(indep)s (enden)m(t)g(set)g(in)e(a)h(graph)g(is)g(a)g(set)h(of)f(pair-)257 2244 y(wise)k(non-adjacen)m(t)f(v)m(ertices)h(and)f(consider)g(the)g Fz(p)-5 b(ar)g(ameterize)g(d)43 b(indep)-5 b(endent)43 b(set)257 2364 y(pr)-5 b(oblem)7 b FB(:)p 408 2467 2936 4 v 408 3028 4 561 v 459 2583 a Fz(p-)p Ft(Independent-Set)746 2704 y Fz(Instanc)-5 b(e:)104 b FB(A)32 b(graph)h Fy(G)f FB(and)h Fy(k)e Fw(2)d Fs(N)9 b FB(.)655 2824 y Fz(Par)-5 b(ameter:)105 b Fy(k)s FB(.)753 2944 y Fz(Pr)-5 b(oblem:)105 b FB(Decide)33 b(if)e Fy(G)i FB(has)g(an)f(indep)s(enden)m(t)i(set)f (of)f(size)h Fy(k)s FB(.)p 3341 3028 V 408 3031 2936 4 v 404 3193 a(Then)26 b Fz(p-)p Ft(Independent-Set)g Fw(\024)1685 3157 y Fr(fpt)1806 3193 y Fz(p-)o Ft(WSa)-8 b(t)o Fx(\(2)p FB(-CNF)o Fx(\))p FB(,)27 b(where)f Fx(2)p FB(-CNF)f(denotes)257 3313 y(the)33 b(class)g(of)f(all)f(prop)s (ositional)e(form)m(ulas)i(in)h(2-conjunctiv)m(e)g(normal)f(form.)404 3434 y(T)-8 b(o)46 b(see)h(this,)i(let)c Fy(G)50 b Fx(=)g(\()p Fy(V)5 b(;)17 b(E)6 b Fx(\))46 b FB(b)s(e)g(a)g(graph.)83 b(F)-8 b(or)45 b(ev)m(ery)j(v)m(ertex)g Fy(v)54 b Fw(2)d Fy(V)67 b FB(w)m(e)257 3554 y(in)m(tro)s(duce)33 b(a)f(prop)s (ositional)d(v)-5 b(ariable)31 b Fy(X)1807 3569 y Fv(v)1880 3554 y FB(whose)j(in)m(tended)f(meaning)e(is)h(\020)8 b Fy(v)36 b FB(b)s(elongs)257 3675 y(to)d(the)g(indep)s(enden)m(t)g (set\021.)44 b(W)-8 b(e)33 b(let)1339 3875 y Fy(\015)g Fx(=)1612 3780 y Fp(^)1526 3996 y Fo(f)p Fv(v)r(;w)r Fo(g2)p Fv(E)1808 3875 y Fx(\()p Fw(:)p Fy(X)1993 3890 y Fv(v)2056 3875 y Fw(_)23 b(:)p Fy(X)2292 3890 y Fv(w)2349 3875 y Fx(\))p Fy(:)257 4168 y FB(Then)45 b Fy(\013)g FB(is)e Fy(k)s FB(-satis\034able)g(if)g(and)g(only)h(if)e Fy(G)i FB(has)g(an)g(indep)s(enden)m(t)h(set)g(of)e(size)h Fy(k)s FB(.)257 4289 y(\(There)32 b(is)f(one)g(detail)e(here)j(that)e (requires)i(atten)m(tion:)42 b(If)31 b Fy(v)j FB(is)d(an)g(isolated)e (v)m(ertex)k(of)257 4409 y Fy(G)p FB(,)c(then)g(the)g(v)-5 b(ariable)26 b Fy(X)1214 4424 y Fv(v)1283 4409 y FB(do)s(es)i(not)g(o)s (ccur)h(in)e Fy(\015)5 b FB(.)42 b(Th)m(us)30 b(the)e(claimed)e(equiv) -5 b(alence)29 b(is)257 4530 y(true)i(for)e(graphs)i(without)e (isolated)g(v)m(ertices.)44 b(W)-8 b(e)30 b(lea)m(v)m(e)h(it)e(to)g (the)i(reader)f(to)g(reduce)257 4650 y(the)j(problem)f(for)g(arbitrary) f(graphs)i(to)f(graphs)h(without)f(isolated)f(v)m(ertices.\))404 4770 y(The)i(con)m(v)m(erse)i(also)d(holds,)g(that)g(is,)922 4959 y Fz(p-)o Ft(WSa)-8 b(t)o Fx(\(2)p FB(-CNF)o Fx(\))28 b Fw(\024)1754 4918 y Fr(fpt)1874 4959 y Fz(p-)p Ft(Independent-Set)o Fy(;)257 5147 y FB(but)k(is)e(m)m(uc)m(h)i(harder)g(to)e(pro)m(v)m(e)j ([)p FA(?)p FB(].)43 b(By)32 b(rev)m(ersing)g(the)f(argumen)m(t)g(ab)s (o)m(v)m(e,)h(it)e(is)h(easy)257 5268 y(to)37 b(sho)m(w)g(that)g Fz(p-)p Ft(WSa)-8 b(t)n Fx(\(2)p FB(-CNF)1531 5225 y Fo(\000)1590 5268 y Fx(\))35 b Fw(\024)1740 5232 y Fr(fpt)1868 5268 y Fz(p-)o Ft(Independent-Set)o FB(,)j(where)g Fx(2)p FB(-CNF)3436 5225 y Fo(\000)257 5388 y FB(denotes)30 b(the)f(class)g(of)f(all)e Fx(2)p FB(-CNF-form)m(ulas)g(in)i(whic)m(h)h (only)f(negativ)m(e)h(literals)d(o)s(ccur.)p eop %%Page: 9 9 9 8 bop 404 573 a FB(W)-8 b(e)33 b(also)e(need)j(a)e(notion)f(of)i (parameterized)e(T)-8 b(uring)32 b(reductions:)257 784 y FA(De\034nition)k(3.8.)49 b FB(Let)31 b Fx(\()p Fy(Q;)17 b(\024)p Fx(\))30 b FB(and)g Fx(\()p Fy(Q)1761 748 y Fo(0)1785 784 y Fy(;)17 b(\024)1885 748 y Fo(0)1908 784 y Fx(\))30 b FB(b)s(e)g(parameterized)g(problems)f(o)m(v)m(er)j(the)257 905 y(alphab)s(ets)42 b Fx(\006)h FB(and)g Fx(\006)1092 868 y Fo(0)1116 905 y FB(,)i(resp)s(ectiv)m(ely)-8 b(.)74 b(An)42 b Fz(fpt)i(T)-7 b(uring)43 b(r)-5 b(e)g(duction)50 b FB(from)41 b Fx(\()p Fy(Q;)17 b(\024)p Fx(\))43 b FB(to)257 1025 y Fx(\()p Fy(Q)372 989 y Fo(0)396 1025 y Fy(;)17 b(\024)496 989 y Fo(0)519 1025 y Fx(\))32 b FB(is)h(an)f(algorithm)d Fs(A)61 b FB(with)32 b(an)g(oracle)g(to)g Fy(Q)2207 989 y Fo(0)2263 1025 y FB(suc)m(h)i(that:)299 1185 y(\(1\))48 b Fs(A)60 b FB(decides)34 b Fx(\()p Fy(Q;)17 b(\024)p Fx(\))p FB(.)299 1342 y(\(2\))48 b Fs(A)60 b FB(is)33 b(an)f(fpt-algorithm)d(with)j(resp)s(ect)i(to)e Fy(\024)p FB(.)299 1500 y(\(3\))48 b(There)f(is)f(a)f(computable)g(function)h Fy(g)53 b Fx(:)e Fs(N)66 b Fw(!)50 b Fs(N)61 b FB(suc)m(h)47 b(that)f(for)f(all)f(oracle)472 1620 y(queries)33 b(\020)8 b Fy(y)31 b Fw(2)d Fy(Q)1091 1584 y Fo(0)1114 1620 y FB(?\021)51 b(p)s(osed)34 b(b)m(y)f Fs(A)61 b FB(on)32 b(input)g Fy(x)h FB(w)m(e)h(ha)m(v)m(e)f Fy(\024)2666 1584 y Fo(0)2690 1620 y Fx(\()p Fy(y)t Fx(\))27 b Fw(\024)h Fy(g)t Fx(\()p Fy(\024)p Fx(\()p Fy(x)p Fx(\)\))p FB(.)257 1832 y(W)-8 b(e)34 b(write)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))28 b Fw(\024)1034 1796 y Fr(fpt-T)1230 1832 y Fx(\()p Fy(Q)1345 1796 y Fo(0)1369 1832 y Fy(;)17 b(\024)1469 1796 y Fo(0)1492 1832 y Fx(\))33 b FB(if)f(there)i(is)f(an)g(fptT)-8 b(uring)32 b(reduction)h(from)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))257 1952 y FB(to)43 b Fx(\()p Fy(Q)502 1916 y Fo(0)526 1952 y Fy(;)17 b(\024)626 1916 y Fo(0)649 1952 y Fx(\))p FB(,)46 b(and)d(w)m(e)h(write)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))46 b Fw(\021)1750 1916 y Fr(fpt-T)1963 1952 y Fx(\()p Fy(Q)2078 1916 y Fo(0)2102 1952 y Fy(;)17 b(\024)2202 1916 y Fo(0)2225 1952 y Fx(\))43 b FB(if)f Fx(\()p Fy(Q;)17 b(\024)p Fx(\))46 b Fw(\024)2782 1916 y Fr(fpt-T)2995 1952 y Fx(\()p Fy(Q)3110 1916 y Fo(0)3134 1952 y Fy(;)17 b(\024)3234 1916 y Fo(0)3257 1952 y Fx(\))43 b FB(and)257 2073 y Fx(\()p Fy(Q)372 2036 y Fo(0)396 2073 y Fy(;)17 b(\024)496 2036 y Fo(0)519 2073 y Fx(\))28 b Fw(\024)662 2036 y Fr(fpt-T)857 2073 y Fx(\()p Fy(Q;)17 b(\024)p Fx(\))p FB(.)257 2313 y FA(3.3)8 b(.)46 b(The)37 b(W-Hierarc)m(h)m(y) 257 2494 y FB(Recall)31 b(the)i(de\034nitions)f(of)g(the)h(classes)h Fx(\000)1842 2509 y Fv(t;d)1960 2494 y FB(of)e(prop)s(ositional)d(form) m(ulas.)257 2705 y FA(De\034nition)36 b(3.9.)91 b FB(\(1\))48 b(F)-8 b(or)32 b Fy(t)c Fw(\025)g Fx(1)p FB(,)33 b(W)p Fx([)p Fy(t)p Fx(])g FB(is)f(the)h(class)g(of)f(all)f(parameterized)h (prob-)472 2826 y(lems)45 b(fpt-reducible)g(to)g(a)h(problem)e Fz(p-)p Ft(WSa)-8 b(t)n Fx(\(\000)2374 2841 y Fv(t;d)2460 2826 y Fx(\))45 b FB(for)g(some)h Fy(d)k Fw(\025)h Fx(1)p FB(,)d(that)472 2946 y(is,)1358 3078 y(W)p Fx([)p Fy(t)p Fx(])28 b(=)1686 2984 y Fp([)1679 3196 y Fv(d)p Fo(\025)p Fu(1)1822 2998 y Fp(\002)1863 3078 y Fz(p-)p Ft(WSa)-8 b(t)n Fx(\(\000)2325 3093 y Fv(t;d)2411 3078 y Fx(\))2449 2998 y Fp(\003)2490 3020 y Fr(fpt)2582 3078 y Fy(:)299 3366 y FB(\(2\))48 b(W)p Fx([)p FB(SA)-8 b(T)q Fx(])32 b FB(is)h(the)g(class)f(of)g(all)f(parameterized)h(problems)f (fpt-reducible)h(to)472 3487 y Fz(p-)p Ft(WSa)-8 b(t)n Fx(\()p FB(PR)m(OP)q Fx(\))o FB(,)33 b(that)f(is,)1287 3709 y(W)p Fx([)p FB(SA)-8 b(T)p Fx(])28 b(=)1762 3628 y Fp(\002)1803 3709 y Fz(p-)p Ft(WSa)-8 b(t)n Fx(\()p FB(PR)m(OP)q Fx(\))2520 3628 y Fp(\003)2561 3650 y Fr(fpt)2654 3709 y Fy(:)299 3932 y FB(\(3\))48 b(W)p Fx([)p FB(P)q Fx(])32 b FB(is)g(the)h(class)g(of)f(all)e(parameterized)i(problems)g (fpt-reducible)g(to)472 4053 y Fz(p-)p Ft(WSa)-8 b(t)n Fx(\()p FB(CIR)m(C)q Fx(\))p FB(,)33 b(that)f(is,)1364 4275 y(W)q Fx([)p FB(P)p Fx(])c(=)1716 4194 y Fp(\002)1758 4275 y Fz(p-)p Ft(WSa)-8 b(t)n Fx(\()p FB(CIR)m(C)q Fx(\))2442 4194 y Fp(\003)2483 4217 y Fr(fpt)2576 4275 y Fy(:)257 4486 y FB(Observ)m(e)35 b(that)887 4691 y(FPT)28 b Fw(\022)g FB(W)p Fx([1])g Fw(\022)g FB(W)p Fx([2])g Fw(\022)g(\001)17 b(\001)g(\001)26 b(\022)i FB(W)p Fx([)p FB(SA)-8 b(T)q Fx(])28 b Fw(\022)g FB(W)p Fx([)p FB(P)q Fx(])p Fy(:)257 4897 y FB(One)c(of)e(the)i(fundamen)m(tal)e(structural)h(results)g(of)g (parameterized)f(complexit)m(y)h(theory)257 5017 y(is)34 b(the)h(follo)m(wing)d(normalization)e(theorem)k(for)g(the)h(W-hierarc) m(h)m(y)-8 b(.)50 b(F)-8 b(or)33 b Fy(t;)17 b(d)31 b Fw(\025)g Fx(1)j FB(w)m(e)257 5137 y(let)d Fx(\000)458 5096 y Fu(+)458 5165 y Fv(t;d)576 5137 y FB(b)s(e)h(the)g(class)g(of)f Fx(\000)1274 5152 y Fv(t;d)1359 5137 y FB(-form)m(ulas)f(in)h(whic)m(h) h(all)e(literals)f(are)j(p)s(ositiv)m(e)f(\(that)h(is,)257 5268 y(no)d(negation)f(sym)m(b)s(ols)h(o)s(ccur\))g(and)g Fx(\000)1694 5227 y Fo(\000)1694 5295 y Fv(t;d)1809 5268 y FB(b)s(e)g(the)g(class)g(of)g Fx(\000)2496 5283 y Fv(t;d)2581 5268 y FB(-form)m(ulas)f(in)g(whic)m(h)h(all)257 5388 y(literals)i(are)h(negativ)m(e)p eop %%Page: 10 10 10 9 bop 257 573 a FA(Theorem)37 b(3.10)h(\(Do)m(wney)f(and)h(F)-9 b(ello)m(ws)36 b([?,)i(?]\).)294 729 y Fz(\(1\))48 b FB(W)p Fx([1])28 b(=)806 648 y Fp(\002)848 729 y Ft(WSa)-8 b(t)n Fx(\(\000)1225 687 y Fo(\000)1225 753 y Fu(1)p Fv(;)p Fu(2)1319 729 y Fx(\))1357 648 y Fp(\003)1399 670 y Fr(fpt)1491 729 y Fy(:)294 922 y Fz(\(2\))48 b(F)-7 b(or)34 b(even)g Fy(t)28 b Fw(\025)g Fx(2)p Fz(,)134 b FB(W)p Fx([)p Fy(t)p Fx(])28 b(=)1580 841 y Fp(\002)1621 922 y Ft(WSa)-8 b(t)o Fx(\(\000)1999 880 y Fu(+)1999 946 y Fv(t;)p Fu(1)2083 922 y Fx(\))2121 841 y Fp(\003)2163 863 y Fr(fpt)2255 922 y Fy(:)294 1115 y Fz(\(3\))48 b(F)-7 b(or)34 b(o)-5 b(dd)34 b Fy(t)28 b Fw(\025)g Fx(3)p Fz(,)134 b FB(W)q Fx([)p Fy(t)p Fx(])28 b(=)1535 1034 y Fp(\002)1577 1115 y Ft(WSa)-8 b(t)n Fx(\(\000)1954 1074 y Fo(\000)1954 1139 y Fv(t;)p Fu(1)2038 1115 y Fx(\))2076 1034 y Fp(\003)2118 1056 y Fr(fpt)2210 1115 y Fy(:)404 1349 y FB(Man)m(y)25 b(natural)f(parameterized)g(problems)f(are)i(complete)f(for)g(the)h (\034rst)g(t)m(w)m(o)g(lev)m(els)257 1470 y(of)j(the)g(W-hierarc)m(h)m (y)-8 b(.)43 b(F)-8 b(or)27 b(example,)h Fz(p-)p Ft(Independent-Set)f FB(is)g(complete)h(for)f(W)p Fx([1])257 1590 y FB([)p FA(?)p FB(],)33 b(and)g(the)g(parameterized)f(dominating)d(set)34 b(problem)d(is)h(complete)g(for)g(W)p Fx([2])g FB([)p FA(?)p FB(].)257 1832 y FA(3.4)8 b(.)46 b(W)p Fx([)p FA(P)o Fx(])38 b FA(and)g(Limited)d(Nondeterminism)257 2013 y FB(W)-8 b(e)33 b(close)f(this)f(in)m(tro)s(ductory)h(section)g (b)m(y)h(presen)m(ting)f(t)m(w)m(o)h(results)f(that)g(establish)f(a)257 2133 y(v)m(ery)26 b(clean)e(connection)h(b)s(et)m(w)m(een)h(the)f (class)f(W)p Fx([)p FB(P)q 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b(tec)m(hniques)h(used)g(to)e(pro)m(v)m (e)h(this)f(result)g(are)g(similar)d(to)j(those)h(in)m(tro)s(duced)f (in)257 693 y(the)33 b(next)g(section.)43 b(Indeed,)34 b(the)e(direction)f(\(1\))d Fx(=)-17 b Fw(\))27 b FB(\(2\))32 b(is)g(an)g(easy)h(consequence)i(of)257 814 y(Theorem)e FA(??)p FB(.)404 941 y(The)c(connection)f(b)s(et)m(w)m(een)j (parameterized)d(complexit)m(y)f(and)i(limited)c(nondeter-)257 1061 y(minism)30 b(can)i(b)s(e)g(broadened)h(if)e(one)h(considers)h Fz(b)-5 b(ounde)g(d)33 b(p)-5 b(ar)g(ameterize)g(d)33 b(c)-5 b(omplexity)257 1182 y(the)g(ory)p FB(,)46 b(where)e(some)e(b)s (ound)h(is)f(put)h(on)g(the)g(gro)m(wth)g(of)f(the)i(dep)s(endence)h (of)d(the)257 1302 y(running)32 b(time)g(of)g(an)g(fpt-algorithm)d(on)j (the)h(parameter)f(\(see)i([)p FA(?)p FB(]\).)257 1699 y FC(4)8 b(.)53 b(The)44 b(M-Hierarc)l(h)l(y)257 1946 y FA(4.1)8 b(.)46 b(A)37 b(New)g(P)m(arameterization)e(of)i(the)h (Satis\034abilit)m(y)c(Problem)257 2134 y FB(In)h(the)g(follo)m(wing,)d 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Fx(\()p Fy(m)p Fx(\))c Fw(2)g Fx(\002\(log)17 b Fy(m)p Fx(\))p Fy(:)404 5268 y FB(These)36 b(considerations)f(motiv)-5 b(ate)33 b(us)j(to)e(in)m(tro)s(duce)h(the)h(follo)m(wing)c(parameteri-)257 5388 y(zation)g(of)g(the)h(satis\034abilit)m(y)d(problem)i(for)g(ev)m (ery)i(class)f Fx(\000)f FB(of)g(circuits.)p eop %%Page: 12 12 12 11 bop 645 477 2464 4 v 645 1102 4 626 v 695 599 a Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)o Fx(\(\000\))982 719 y Fz(Instanc)j(e:)104 b Fy(\015)33 b Fw(2)28 b Fx(\000)k FB(of)g(size)h Fy(m)g FB(with)f Fy(n)h FB(v)-5 b(ariables.)891 870 y Fz(Par)g(ameter:)1471 759 y Fp(l)1595 830 y Fv(n)p 1534 847 166 4 v 1534 904 a Fu(log)12 b Fv(m)1709 759 y Fp(m)1761 870 y FB(.)990 1019 y Fz(Pr)-5 b(oblem:)104 b FB(Decide)33 b(if)f Fy(\015)37 b FB(is)32 b(satis\034able.)p 3105 1102 4 626 v 645 1105 2464 4 v 257 1295 a(Ob)m(viously)-8 b(,)33 b Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)n Fx(\(\000\))33 b FB(is)f(solv)-5 b(able)31 b(in)h(time)1073 1513 y Fx(2)1122 1472 y Fv(n)1191 1513 y Fw(\001)22 b Fy(m)1326 1472 y Fv(O)r Fu(\(1\))1504 1513 y Fw(\024)28 b Fx(2)1658 1472 y Fv(k)r Fo(\001)p Fu(log)12 b Fv(m)1907 1513 y Fw(\001)22 b Fy(m)2042 1472 y Fv(O)r Fu(\(1\))2220 1513 y Fx(=)28 b Fy(m)2409 1472 y Fv(k)r Fu(+)p Fv(O)r Fu(\(1\))2652 1513 y Fy(;)257 1754 y FB(where)42 b Fy(k)h Fx(=)758 1643 y Fp(l)882 1714 y Fv(n)p 820 1731 166 4 v 820 1788 a Fu(log)13 b Fv(m)996 1643 y Fp(m)1088 1754 y FB(is)40 b(the)g(parameter.)66 b(In)m(tuitiv)m(ely)40 b(it)f(seems)i(unlik)m(ely)e(that)h(the)257 1895 y(problem)32 b(is)g(\034xed-parameter)g(tractable.)404 2016 y(T)-8 b(o)31 b(phrase)h(our)f(\034rst)g(result)g(in)f(its)h(most) f(general)h(form,)f(w)m(e)i(in)m(tro)s(duce)f(a)g(simple)257 2136 y(closure)h(prop)s(ert)m(y)h(of)e(classes)i(of)e(circuits:)42 b(W)-8 b(e)33 b(call)d(a)h(class)h Fx(\000)g Fz(p)-5 b(addable)38 b FB(if)30 b(for)h(ev)m(ery)257 2256 y Fy(\015)43 b Fw(2)38 b Fx(\000)g FB(and)g(for)g(ev)m(ery)i Fy(m)1252 2220 y Fo(0)1313 2256 y Fw(\025)e(j)p Fy(\015)5 b Fw(j)37 b FB(there)i(is)f(a)g(circuit)f Fy(\015)2388 2220 y Fo(0)2448 2256 y Fw(2)h Fx(\000)g FB(suc)m(h)i(that)e Fx(v)-5 b(ar\()p Fy(\015)3321 2220 y Fo(0)3344 2256 y Fx(\))37 b(=)257 2377 y(v)-5 b(ar\()p Fy(\015)5 b Fx(\))p FB(,)27 b(the)f(circuits)f Fy(\015)30 b FB(and)c Fy(\015)1392 2341 y Fo(0)1440 2377 y FB(are)g(equiv)-5 b(alen)m(t,)26 b(and)g Fy(m)2348 2341 y Fo(0)2399 2377 y Fw(\024)i(j)p Fy(\015)2588 2341 y Fo(0)2611 2377 y Fw(j)g(\024)g Fy(O)s Fx(\()p Fy(m)2973 2341 y Fo(0)2996 2377 y Fx(\))p FB(.)41 b(W)-8 b(e)26 b(call)d Fx(\000)257 2497 y Fz(e\036ciently)31 b(p)-5 b(addable)34 b FB(if,)29 b(in)e(addition,)h(there)h(is)f(an)g (algorithm)d(that)j(computes)h Fy(\015)3327 2461 y Fo(0)3379 2497 y FB(for)257 2618 y(giv)m(en)34 b Fy(\015)k FB(and)c Fy(m)878 2581 y Fo(0)931 2618 y Fw(\025)29 b(j)p Fy(\015)5 b Fw(j)33 b FB(in)g(time)f Fx(\()p Fy(m)1643 2581 y Fo(0)1667 2618 y Fx(\))1705 2581 y Fv(O)r Fu(\(1\))1854 2618 y FB(.)47 b(Most)33 b(natural)g(classes)h(of)f(form)m(ulas)f(and)257 2738 y(circuits)j(are)f(e\036cien)m(tly)i(paddable,)f(in)f(particular)f (all)g(classes)j Fx(\000)2735 2753 y Fv(t;d)2855 2738 y FB(and)f(the)g(classes)257 2858 y(PR)m(OP)f(and)e(CIR)m(C)q(.)44 b(F)-8 b(or)31 b(example,)h(for)g(the)h Fx(\000)2034 2873 y Fu(1)p Fv(;)p Fu(2)2129 2858 y FB(-form)m(ula)1498 3156 y Fy(\015)g Fx(=)1712 3031 y Fv(m)1687 3061 y Fp(^)1686 3271 y Fv(i)p Fu(=1)1800 3156 y Fx(\()p Fy(\025)1895 3171 y Fv(i)p Fu(1)1980 3156 y Fw(_)23 b Fy(\025)2126 3171 y Fv(i)p Fu(2)2189 3156 y Fx(\))p Fy(;)257 3461 y FB(w)m(e)34 b(can)f(let)f Fy(\025)778 3476 y Fv(ij)866 3461 y Fx(=)27 b Fy(\025)1026 3476 y Fv(mj)1158 3461 y FB(for)32 b Fy(m)c(<)f(i)h Fw(\024)g Fy(m)1774 3424 y Fo(0)1830 3461 y FB(and)33 b Fy(j)h Fx(=)27 b(1)p Fy(;)17 b Fx(2)p FB(,)32 b(and)1487 3781 y Fy(\015)1543 3739 y Fo(0)1594 3781 y Fx(=)1712 3656 y Fv(m)1774 3633 y Fk(0)1699 3686 y Fp(^)1697 3896 y Fv(i)p Fu(=1)1811 3781 y Fx(\()p Fy(\025)1906 3796 y Fv(i)p Fu(1)1992 3781 y Fw(_)22 b Fy(\025)2137 3796 y Fv(i)p Fu(2)2201 3781 y Fx(\))p Fy(:)257 4095 y FA(Prop)s(osition)36 b(4.1)h(\([?,)g(?]\).)49 b Fz(L)-5 b(et)35 b Fx(\000)g Fz(b)-5 b(e)34 b(an)g(e\036ciently)h(p)-5 b(addable)33 b(class)h(of)g(cir)-5 b(cuits.)257 4216 y(Then)575 4434 y(p-lo)g(g-)o Ft(Sa)d(t)o Fx(\(\000\))27 b Fw(2)h FB(FPT)56 b Fw(\()-17 b(\))55 b Ft(Sa)-8 b(t)o Fx(\(\000\))27 b Fw(2)h FB(DTIME)2499 4353 y Fp(\000)2545 4434 y Fx(2)2594 4393 y Fv(o)2628 4370 y Fq(e\033)2696 4393 y Fu(\()p Fv(n)p Fu(\))2820 4434 y Fw(\001)22 b Fy(m)2955 4393 y Fv(O)r Fu(\(1\))3105 4353 y Fp(\001)3150 4434 y Fy(;)257 4653 y Fz(wher)-5 b(e)42 b Fy(n)f Fx(=)g Fw(j)p Fx(v)-5 b(ar\()p Fy(\015)5 b Fx(\))p Fw(j)41 b Fz(is)h(the)g(numb)-5 b(er)42 b(of)g(variables)e(and)i Fy(m)f Fx(=)g Fw(j)p Fy(\015)5 b Fw(j)42 b Fz(the)g(size)f(of)h(the)257 4773 y(input)35 b(cir)-5 b(cuit)35 b Fy(\015)5 b Fz(.)257 5049 y(Pr)-5 b(o)g(of:)62 b FB(Supp)s(ose)39 b(\034rst)g(that)g Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)n Fx(\(\000\))38 b Fw(2)g FB(FPT)q(.)62 b(Let)38 b Fy(f)49 b Fx(:)38 b Fs(N)53 b Fw(!)38 b Fs(N)53 b FB(b)s(e)39 b(a)f(com-)257 5170 y(putable)33 b(function)f(and)g Fs(A)61 b FB(an)32 b(fpt-algorithm)d (that)j(decides)i Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)n Fx(\(\000\))33 b FB(in)e(time)1615 5388 y Fy(f)11 b Fx(\()p Fy(k)s Fx(\))22 b Fw(\001)f Fy(m)1960 5347 y Fv(O)r Fu(\(1\))2111 5388 y Fy(;)p eop %%Page: 13 13 13 12 bop 257 573 a FB(where)37 b Fy(k)f Fx(=)d Fw(d)p Fy(n=)17 b Fx(log)f Fy(m)p Fw(e)37 b FB(is)e(the)h(parameter.)52 b(Without)35 b(loss)g(of)g(generalit)m(y)-8 b(,)36 b(w)m(e)g(ma)m(y)257 693 y(assume)27 b(that)f Fy(f)37 b FB(is)25 b(increasing)h(and)g(time)f (constructible,)i(whic)m(h)g(implies)c(that)j Fy(f)11 b Fx(\()p Fy(i)p Fx(\))28 b Fw(\024)257 814 y Fy(j)39 b FB(can)33 b(b)s(e)f(decided)i(in)d(time)g Fy(O)s Fx(\()p Fy(j)6 b Fx(\))p FB(.)43 b(Let)33 b Fy(\023)28 b Fx(:)g Fs(N)42 b Fw(!)28 b Fs(N)47 b FB(b)s(e)33 b(de\034ned)h(b)m(y)1040 1029 y Fy(\023)p Fx(\()p Fy(n)p Fx(\))29 b(=)e(max)1538 919 y Fp(\020)1598 1029 y Fw(f)p Fx(1)p Fw(g)21 b([)i(f)p Fy(i)28 b Fw(2)g Fs(N)42 b Fw(j)28 b Fy(f)11 b Fx(\()p Fy(i)p Fx(\))27 b Fw(\024)i Fy(n)p Fw(g)2626 919 y Fp(\021)2685 1029 y Fy(:)257 1245 y FB(Then)45 b Fy(\023)f FB(is)f(non-decreasing)h (and)f(un)m(b)s(ounded,)48 b Fy(f)11 b Fx(\()p Fy(\023)p Fx(\()p Fy(n)p Fx(\)\))47 b Fw(\024)g Fy(n)d FB(for)f(all)e(but)j (\034nitely)257 1365 y(man)m(y)33 b Fy(n)p FB(,)g(and)f Fy(\023)p Fx(\()p Fy(n)p Fx(\))h FB(can)g(b)s(e)g(computed)g(in)e(time) h Fy(O)s Fx(\()p Fy(n)2303 1329 y Fu(2)2341 1365 y Fx(\))p FB(.)404 1486 y(W)-8 b(e)33 b(shall)e(pro)m(v)m(e)j(that)e Ft(Sa)-8 b(t)n Fx(\(\000\))28 b Fw(2)g FB(DTIME)2040 1405 y Fp(\000)2086 1486 y Fx(2)2135 1450 y Fv(O)r Fu(\()p Fv(n=\023)p Fu(\()p Fv(n)p Fu(\)\))2472 1486 y Fw(\001)22 b Fy(m)2607 1450 y Fv(O)r Fu(\(1\))2757 1405 y Fp(\001)2802 1486 y FB(.)404 1615 y(Let)28 b Fy(\015)33 b Fw(2)28 b Fx(\000)p FB(,)h Fy(m)f Fx(=)g Fw(j)p Fy(\015)5 b Fw(j)28 b FB(and)g Fy(n)g Fx(=)g Fw(j)p Fx(v)-5 b(ar)o(\()p Fy(\015)5 b Fx(\))p Fw(j)p FB(.)42 b(Assume)29 b(\034rst)g(that)f Fy(m)g Fw(\025)g Fx(2)3018 1579 y Fv(n=\023)p Fu(\()p Fv(n)p Fu(\))3223 1615 y FB(.)42 b(Note)257 1735 y(that)1437 1888 y Fy(k)31 b Fx(=)1623 1747 y Fp(\030)1776 1821 y Fy(n)p 1691 1865 228 4 v 1691 1956 a Fx(log)16 b Fy(m)1929 1747 y Fp(\031)2014 1888 y Fw(\024)29 b Fy(\023)p Fx(\()p Fy(n)p Fx(\))p Fy(:)257 2106 y FB(Th)m(us)40 b Fy(f)11 b Fx(\()p Fy(k)s Fx(\))38 b Fw(\024)g Fy(f)11 b Fx(\()p Fy(\023)p Fx(\()p Fy(n)p Fx(\)\))38 b Fw(\024)h Fy(n)p FB(,)h(and)f(w)m(e)g(can)g(simply)e(decide)i Fy(\015)k Fw(2)c Ft(Sa)-8 b(t)n Fx(\(\000\))39 b FB(with)f(the)257 2226 y(fpt-algorithm)29 b Fs(A)61 b FB(in)31 b(time)1183 2415 y Fy(f)11 b Fx(\()p Fy(k)s Fx(\))21 b Fw(\001)h Fy(m)1528 2373 y Fv(O)r Fu(\(1\))1706 2415 y Fw(\024)28 b Fy(n)23 b Fw(\001)e Fy(m)2026 2373 y Fv(O)r Fu(\(1\))2204 2415 y Fx(=)28 b Fy(m)2393 2373 y Fv(O)r Fu(\(1\))2543 2415 y Fy(:)257 2603 y FB(Assume)44 b(next)f(that)g Fy(m)i(<)g Fx(2)1380 2567 y Fv(n=\023)p Fu(\()p Fv(n)p Fu(\))1585 2603 y FB(.)74 b(Let)42 b Fy(m)1955 2567 y Fo(0)2024 2603 y Fx(=)j(2)2194 2567 y Fo(d)p Fv(n=\023)p Fu(\()p Fv(n)p Fu(\))p Fo(e)2461 2603 y FB(.)74 b(Let)43 b Fy(\015)2803 2567 y Fo(0)2871 2603 y Fw(2)j Fx(\000)c FB(suc)m(h)i(that)257 2723 y Fx(v)-5 b(ar\()p Fy(\015)484 2687 y Fo(0)507 2723 y Fx(\))28 b(=)g(v)-5 b(ar)o(\()p Fy(\015)5 b Fx(\))p FB(,)29 b(the)e(circuits)g Fy(\015)32 b FB(and)27 b Fy(\015)1819 2687 y Fo(0)1870 2723 y FB(are)g(equiv)-5 b(alen)m(t,)28 b(and)f Fy(m)2782 2687 y Fo(0)2834 2723 y Fw(\024)h(j)p Fy(\015)3023 2687 y Fo(0)3046 2723 y Fw(j)f(\024)h Fy(O)s Fx(\()p Fy(m)3407 2687 y Fo(0)3430 2723 y Fx(\))p FB(.)257 2844 y(Since)f Fx(\000)g FB(is)f(e\036cien)m(tly)i(paddable,)f(suc)m(h) i(a)d Fy(\015)1905 2807 y Fo(0)1955 2844 y FB(can)h(b)s(e)g(computed)g (in)f(time)g(p)s(olynomial)257 2964 y(in)j Fy(m)453 2928 y Fo(0)476 2964 y FB(,)h(that)f(is,)h(time)d Fx(2)1131 2928 y Fv(O)r Fu(\()p Fv(n=\023)p Fu(\()p Fv(n)p Fu(\)\))1446 2964 y FB(.)43 b(Let)29 b Fy(k)1741 2928 y Fo(0)1792 2964 y Fx(=)f Fy(n=)17 b Fx(log)f Fw(j)p Fy(\015)2246 2928 y Fo(0)2269 2964 y Fw(j)p FB(.)42 b(Then)30 b Fy(k)2671 2928 y Fo(0)2722 2964 y Fw(\024)e Fy(\023)p Fx(\()p Fy(n)p Fx(\))p FB(.)43 b(W)-8 b(e)30 b(decide)257 3084 y Fy(\015)313 3048 y Fo(0)364 3084 y Fw(2)e Ft(Sa)-8 b(t)o Fx(\(\000\))32 b FB(with)h(the)g(fpt-algorithm)28 b Fs(A)61 b FB(in)31 b(time)1240 3273 y Fy(f)11 b Fx(\()p Fy(k)1391 3232 y Fo(0)1414 3273 y Fx(\))22 b Fw(\001)g Fx(\()p Fy(m)1647 3232 y Fo(0)1671 3273 y Fx(\))1709 3232 y Fv(O)r Fu(\(1\))1886 3273 y Fw(\024)28 b Fy(n)23 b Fw(\001)e Fx(2)2170 3232 y Fv(O)r Fu(\()p Fv(n=\023)p Fu(\()p Fv(n)p Fu(\)\))2485 3273 y Fy(:)257 3461 y FB(This)33 b(completes)f(the)h(pro)s(of)f(of)g (the)h(forw)m(ard)g(direction.)404 3622 y(F)-8 b(or)47 b(the)i(bac)m(kw)m(ard)h(direction,)h(let)c Fs(B)70 b FB(b)s(e)48 b(an)g(algorithm)d(solving)i Ft(Sa)-8 b(t)o Fx(\(\000\))48 b FB(in)257 3742 y(DTIME)593 3662 y Fp(\000)639 3742 y Fx(2)688 3706 y Fv(O)r Fu(\()p Fv(n=\023)p Fu(\()p Fv(n)p Fu(\)\))1034 3742 y Fw(\001)30 b Fy(m)1177 3706 y Fv(O)r Fu(\(1\))1327 3662 y Fp(\001)1418 3742 y FB(for)45 b(some)g(computable)f(function)g Fy(\023)50 b Fx(:)f Fs(N)65 b Fw(!)48 b Fs(N)60 b FB(that)257 3863 y(is)43 b(non-decreasing)f(and)h(un)m(b)s(ounded.)76 b(Let)43 b Fy(f)53 b FB(b)s(e)43 b(a)f(non-decreasing)h(computable)257 3983 y(function)32 b(with)g Fy(f)11 b Fx(\()p Fy(\023)p Fx(\()p Fy(n)p Fx(\)\))28 b Fw(\025)g Fx(2)1346 3947 y Fv(n)1426 3983 y FB(for)k(all)e Fy(n)e Fw(2)g Fs(N)9 b FB(.)50 b(W)-8 b(e)33 b(claim)d(that)1077 4171 y Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)o Fx(\(\000\))27 b Fw(2)h FB(DTIME)r Fx(\()p Fy(f)11 b Fx(\()p Fy(k)s Fx(\))22 b Fw(\001)g Fy(m)2461 4130 y Fv(O)r Fu(\(1\))2611 4171 y Fx(\))p Fy(:)257 4360 y FB(Let)38 b Fy(\015)j Fw(2)36 b Fx(\000)p FB(,)i Fy(m)e Fx(=)g Fw(j)p Fy(\015)5 b Fw(j)p FB(,)38 b Fy(n)e Fx(=)f Fw(j)p Fx(v)-5 b(ar\()p Fy(\015)5 b Fx(\))p Fw(j)p FB(,)38 b(and)f Fy(k)i Fx(=)c Fw(d)q Fy(n=)17 b Fx(log)f Fy(m)p Fw(e)q FB(.)57 b(If)37 b Fy(m)f Fw(\025)h Fx(2)3064 4324 y Fv(n=\023)p Fu(\()p Fv(n)p Fu(\))3306 4360 y FB(then)257 4480 y(algorithm)30 b Fs(B)54 b FB(decides)33 b Fy(\015)g Fw(2)28 b Ft(Sa)-8 b(t)n Fx(\(\000\))33 b FB(in)f(time)f Fy(m)2077 4444 y Fv(O)r Fu(\(1\))2227 4480 y FB(.)43 b(If)33 b Fy(m)28 b(<)f Fx(2)2660 4444 y Fv(n=\023)p Fu(\()p Fv(n)p Fu(\))2865 4480 y FB(,)33 b(then)1451 4726 y Fy(k)e Fx(=)1636 4585 y Fp(\030)1789 4658 y Fy(n)p 1704 4703 V 1704 4794 a Fx(log)17 b Fy(m)1942 4585 y Fp(\031)2028 4726 y Fw(\025)28 b Fy(\023)p Fx(\()p Fy(n)p Fx(\))257 4971 y FB(and)37 b(th)m(us)h Fy(f)11 b Fx(\()p Fy(k)s Fx(\))35 b Fw(\025)g Fx(2)1055 4935 y Fv(n)1102 4971 y FB(.)56 b(Th)m(us)39 b(w)m(e)f(can)f(decide)g Fy(\015)j Fw(2)c Ft(Sa)-8 b(t)n Fx(\(\000\))37 b FB(b)m(y)h(exhaustiv)m (e)g(searc)m(h)257 5092 y(in)32 b(time)f Fy(O)s Fx(\()p Fy(f)11 b Fx(\()p Fy(k)s Fx(\))21 b Fw(\001)h Fy(m)p Fx(\))p FB(.)2302 b Fj(2)404 5268 y FB(In)32 b(the)h(follo)m(wing,)c(w) m(e)k(shall)e(sa)m(y)i(that)f Ft(Sa)-8 b(t)o Fx(\(\000\))32 b FB(is)f Fz(sub)-5 b(exp)g(onential)33 b(\(with)i(r)-5 b(esp)g(e)g(ct)257 5388 y(to)35 b(the)g(numb)-5 b(er)35 b(of)f(variables\))h FB(if)c(it)h(is)g(solv)-5 b(able)31 b(in)h(DTIME)2545 5307 y Fp(\000)2591 5388 y Fx(2)2640 5352 y Fv(o)2674 5329 y Fq(e\033)2741 5352 y Fu(\()p Fv(n)p Fu(\))2865 5388 y Fw(\001)22 b Fy(m)3000 5352 y Fv(O)r Fu(\(1\))3150 5307 y Fp(\001)3196 5388 y FB(.)p eop %%Page: 14 14 14 13 bop 257 573 a FA(4.2)8 b(.)46 b(The)37 b(M-Hierarc)m(h)m(y)257 755 y FB(Motiv)-5 b(ated)35 b(b)m(y)h(Prop)s(osition)e FA(??)p FB(,)i(w)m(e)g(de\034ne)h(another)e(hierarc)m(h)m(y)h(of)f (parameterized)257 875 y(complexit)m(y)d(classes)i(in)d(anlogy)h(to)g (De\034nition)g(3.8:)257 1124 y FA(De\034nition)k(4.2.)91 b FB(\(1\))48 b(F)-8 b(or)21 b(ev)m(ery)i Fy(t)28 b Fw(\025)g Fx(1)p FB(,)c(w)m(e)f(let)e(M)p Fx([)p Fy(t)p Fx(])28 b(=)2469 1049 y Fp(S)2552 1153 y Fv(d)p Fo(\025)p Fu(1)2699 1043 y Fp(\002)2741 1124 y Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)n Fx(\(\000)3245 1139 y Fv(t;d)3331 1124 y Fx(\))3369 1043 y Fp(\003)3410 1066 y Fr(fpt)3503 1124 y Fy(:)299 1321 y FB(\(2\))48 b(M)p Fx([)p FB(SA)-8 b(T)q Fx(])28 b(=)936 1240 y Fp(\002)978 1321 y Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)o Fx(\()p FB(PR)m(OP)p Fx(\))1737 1240 y Fp(\003)1779 1262 y Fr(fpt)1871 1321 y FB(.)299 1512 y(\(3\))48 b(M)p Fx([)p FB(P)q Fx(])28 b(=)813 1431 y Fp(\002)855 1512 y Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)n Fx(\()p FB(CIR)m(C)q Fx(\))1581 1431 y Fp(\003)1623 1454 y Fr(fpt)1715 1512 y FB(.)257 1749 y(Then)34 b(b)m(y)f(Prop)s(osition)e FA(??)p FB(:)257 1987 y FA(Corollary)36 b(4.3.)86 b Fz(\(1\))48 b(F)-7 b(or)47 b Fy(t)52 b Fw(\025)g Fx(1)p Fz(,)f FB(M)q Fx([)p Fy(t)p Fx(])h(=)g FB(FPT)c Fz(if)g(and)f(only)h(if)g Ft(Sa)-8 b(t)n Fx(\(\000)3254 2002 y Fv(t;d)3340 1987 y Fx(\))48 b Fz(is)472 2107 y(sub)-5 b(exp)g(onential)33 b(for)i(al)5 b(l)34 b Fy(d)28 b Fw(\025)g Fx(1)p Fz(.)294 2272 y(\(2\))48 b FB(M)p Fx([)p FB(SA)-8 b(T)q Fx(])28 b(=)f FB(FPT)36 b Fz(if)e(and)h(only)f(if)h Ft(Sa)-8 b(t)n Fx(\()p FB(PR)m(OP)q Fx(\))35 b Fz(is)f(sub)-5 b(exp)g(onential.)294 2436 y(\(3\))48 b FB(M)p Fx([)p FB(P)q Fx(])28 b(=)f FB(FPT)35 b Fz(if)g(and)f(only)h(if)g Ft(Sa)-8 b(t)n Fx(\()p FB(CIR)m(C)q Fx(\))35 b Fz(is)g(sub)-5 b(exp)g(onential.)257 2674 y FB(The)38 b(follo)m(wing)33 b(theorem)j(is)g(essen)m(tially)g(due)h(to)e(Abrahamson,)i(Do)m(wney)-8 b(,)39 b(and)e(F)-8 b(el-)257 2794 y(lo)m(ws)33 b([)p FA(?)p FB(])g(\(also)e(see)j([)p FA(?)p FB(]\).)257 3031 y FA(Theorem)j(4.4.)49 b Fz(F)-7 b(or)34 b(every)h Fy(t)28 b Fw(\025)g Fx(1)p Fz(,)1372 3255 y FB(M)p Fx([)p Fy(t)p Fx(])g Fw(\022)g FB(W)q Fx([)p Fy(t)p Fx(])g Fw(\022)g FB(M)p Fx([)p Fy(t)23 b Fx(+)f(1])p Fy(:)257 3479 y Fz(F)-7 b(urthermor)i(e,)34 b FB(M)p Fx([)p FB(SA)-8 b(T)q Fx(])28 b(=)f FB(W)q Fx([)p FB(SA)-8 b(T)p Fx(])35 b Fz(and)g FB(M)p Fx([)p FB(P)p Fx(])28 b(=)g FB(W)p Fx([)p FB(P)p Fx(])p Fz(.)257 3769 y(Pr)-5 b(o)g(of:)64 b FB(W)-8 b(e)41 b(\034rst)g(pro)m(v)m(e)g(M)p Fx([)p Fy(t)p Fx(])h Fw(\022)f FB(W)p Fx([)p Fy(t)p Fx(])p FB(.)67 b(F)-8 b(or)39 b(simplicit)m(y)-8 b(,)39 b(let)h(us)h(assume)f(that)g Fy(t)h FB(is)257 3889 y(o)s(dd.)j(Fix)31 b Fy(d)d Fw(\025)g Fx(1)k FB(suc)m(h)i(that)e Fy(t)23 b Fx(+)f Fy(d)27 b Fw(\025)h Fx(3)p FB(.)44 b(W)-8 b(e)32 b(shall)g(pro)m(v)m(e)h(that)1143 4113 y Fz(p-lo)-5 b(g-)p Ft(Sa)d(t)n Fx(\(\000)1648 4128 y Fv(t;d)1734 4113 y Fx(\))27 b Fw(\024)1876 4072 y Fr(fpt)1997 4113 y Fz(p-)o Ft(WSa)-8 b(t)o Fx(\(\000)2459 4128 y Fv(t;d)2544 4113 y Fx(\))p Fy(:)686 b FB(\(4.1\))257 4337 y(Let)33 b Fy(\015)g Fw(2)28 b Fx(\000)671 4352 y Fv(t;d)756 4337 y FB(.)44 b(W)-8 b(e)33 b(shall)e(construct)i(a)g Fx(\000)1794 4352 y Fv(t;d)1879 4337 y FB(-form)m(ula)d Fy(\014)38 b FB(suc)m(h)c(that)1066 4561 y Fy(\015)j FB(is)32 b(satis\034able)55 b Fw(\()-17 b(\))55 b Fy(\014)38 b FB(is)32 b Fy(k)s FB(-satis\034able)o Fy(:)608 b FB(\(4.2\))257 4785 y(Let)44 b Fy(m)j Fx(=)g Fw(j)p Fy(\015)5 b Fw(j)p FB(,)46 b Fy(n)h Fx(=)f Fw(j)p Fx(v)-5 b(ar\()p Fy(\015)5 b Fx(\))p Fw(j)p FB(.)77 b(T)-8 b(o)43 b(simplify)e(the)k(notation,)g(let)e(us)h(assume) g(that)257 4905 y Fy(`)36 b Fx(=)f(log)16 b Fy(m)38 b FB(and)f Fy(k)h Fx(=)d Fy(n=)17 b Fx(log)f Fy(m)38 b FB(are)f(in)m(tegers.)57 b(Then)38 b Fy(n)d Fx(=)g Fy(k)29 b Fw(\001)c Fy(`)p FB(.)56 b(Let)37 b Fw(X)50 b Fx(=)35 b(v)-5 b(ar\()p Fy(\015)5 b Fx(\))p FB(,)257 5026 y(and)33 b(let)f Fw(X)659 5041 y Fu(1)698 5026 y Fy(;)17 b(:)g(:)g(:)f(;)h Fw(X)988 5041 y Fv(k)1063 5026 y FB(b)s(e)33 b(a)f(partition)e(of)j Fw(X)47 b FB(in)m(to)31 b Fy(k)36 b FB(sets)e(of)e(size)h Fy(`)p FB(.)404 5147 y(F)-8 b(or)30 b Fx(1)e Fw(\024)g Fy(i)g Fw(\024)g Fy(k)35 b FB(and)d(ev)m(ery)h(subset)g Fy(S)h Fw(\022)28 b(X)2024 5162 y Fv(i)2052 5147 y FB(,)k(let)f Fy(Y)2329 5111 y Fv(S)2308 5172 y(i)2412 5147 y FB(b)s(e)g(a)h(new)g(v) -5 b(ariable.)42 b(Let)31 b Fw(Y)3467 5162 y Fv(i)257 5268 y FB(b)s(e)37 b(the)g(set)g(of)g(all)d Fy(Y)1055 5232 y Fv(S)1034 5292 y(i)1143 5268 y FB(and)i Fw(Y)43 b Fx(=)1556 5193 y Fp(S)1639 5219 y Fv(k)1639 5297 y(i)p Fu(=1)1774 5268 y Fw(Y)1841 5283 y Fv(i)1869 5268 y FB(.)55 b(Call)35 b(a)h(truth)h(v)-5 b(alue)36 b(assignmen)m(t)g(for)g Fw(Y)257 5388 y Fz(go)-5 b(o)g(d)47 b FB(if)36 b(for)h Fx(1)f Fw(\024)g Fy(i)g Fw(\024)g Fy(k)k FB(exactly)e(one)g(v)-5 b(ariable)35 b(in)i Fw(Y)2289 5403 y Fv(i)2354 5388 y FB(is)f(set)i(to)f Ft(tr)m(ue)p FB(.)57 b(There)39 b(is)d(a)p eop %%Page: 15 15 15 14 bop 257 573 a FB(bijection)29 b Fy(f)41 b FB(b)s(et)m(w)m(een)32 b(the)f(truth)f(v)-5 b(alue)30 b(assignmen)m(ts)g Fw(V)39 b FB(for)29 b Fw(X)45 b FB(and)30 b(the)h(go)s(o)s(d)e(truth)257 693 y(v)-5 b(alue)32 b(assignmen)m(ts)h(for)f Fw(Y)40 b FB(de\034ned)34 b(b)m(y)474 873 y Fy(f)11 b Fx(\()p Fw(V)d Fx(\)\()p Fy(Y)794 832 y Fv(S)773 898 y(i)844 873 y Fx(\))28 b(=)g Ft(tr)m(ue)54 b Fw(\()-17 b(\))55 b(8)p Fy(X)36 b Fw(2)28 b(X)1879 888 y Fv(i)1935 873 y Fx(:)2018 792 y Fp(\000)2063 873 y Fw(V)8 b Fx(\()p Fy(X)g Fx(\))28 b(=)f Ft(tr)m(ue)55 b Fw(\()-17 b(\))55 b Fy(X)35 b Fw(2)29 b Fy(S)3234 792 y Fp(\001)257 1053 y FB(for)j(all)f Fw(V)36 b Fx(:)28 b Fw(X)42 b(!)27 b(f)p Ft(tr)m(ue)o Fy(;)17 b Ft(f)-11 b(alse)o Fw(g)p FB(,)33 b Fx(1)27 b Fw(\024)h Fy(i)g Fw(\024)g Fy(k)s FB(,)33 b(and)g Fy(S)g Fw(\022)28 b(X)2551 1068 y Fv(i)2580 1053 y FB(.)404 1173 y(Let)37 b Fy(\014)644 1137 y Fo(00)723 1173 y FB(b)s(e)g(the)g(form)m(ula)e(obtained)h(from)g Fy(\015)42 b FB(b)m(y)c(replacing,)e(for)h Fx(1)e Fw(\024)g Fy(i)h Fw(\024)f Fy(k)40 b FB(and)257 1293 y Fy(X)c Fw(2)28 b(X)539 1308 y Fv(i)567 1293 y FB(,)33 b(eac)m(h)g(o)s(ccurrence)h(of)e (the)h(literal)d Fy(X)40 b FB(b)m(y)34 b(the)f(form)m(ula)1715 1390 y Fp(^)1511 1603 y Fv(S)t Fo(\022X)1663 1613 y Fi(i)1712 1603 y Fr(with)24 b Fv(X)5 b Fo(62)p Fv(S)2046 1485 y Fw(:)p Fy(Y)2191 1444 y Fv(S)2169 1510 y(i)257 1762 y FB(and)33 b(eac)m(h)g(o)s(ccurrence)h(of)e(the)h(literal)d Fw(:)p Fy(X)41 b FB(b)m(y)33 b(the)g(form)m(ula)1701 1859 y Fp(^)1497 2071 y Fv(S)t Fo(\022X)1649 2081 y Fi(i)1699 2071 y Fr(with)23 b Fv(X)5 b Fo(2)p Fv(S)2033 1953 y Fw(:)p Fy(Y)2178 1912 y Fv(S)2156 1978 y(i)2228 1953 y Fy(:)257 2234 y FB(Then)37 b(an)e(assignmen)m(t)h Fw(V)41 b Fx(:)32 b Fw(X)48 b(!)32 b(f)p Ft(tr)m(ue)o Fy(;)17 b Ft(f)-11 b(alse)o Fw(g)35 b FB(satis\034es)i Fy(\015)j FB(if)34 b(and)i(only)f(if)f Fy(f)11 b Fx(\()p Fw(V)d Fx(\))257 2355 y FB(satis\034es)41 b Fy(\014)685 2318 y Fo(00)727 2355 y FB(.)64 b(Th)m(us)41 b Fy(\015)k FB(is)39 b(satis\034able)g(if)f(and)i(only)f(if)f Fy(\014)2402 2318 y Fo(00)2484 2355 y FB(has)i(a)f(go)s(o)s(d)f(assignmen)m(t.)257 2475 y(Note)h(that)f(the)h(size)g(of)f(eac)m(h)h(of)f(the)h(sets)g Fw(Y)1977 2490 y Fv(i)2043 2475 y FB(is)f Fx(2)2196 2439 y Fv(`)2267 2475 y Fx(=)f Fy(m)p FB(.)62 b(Th)m(us)40 b(the)f(size)f(of)g Fy(\014)3349 2439 y Fo(00)3430 2475 y FB(is)257 2595 y(p)s(olynomial)24 b(in)i Fy(m)p FB(.)42 b(Moreo)m(v)m(er,)30 b Fy(\014)1530 2559 y Fo(00)1599 2595 y FB(can)d(easily)g(b)s(e)g(computed)g(from)f Fy(\015)32 b FB(in)27 b(p)s(olynomial)257 2716 y(time.)404 2836 y Fy(\014)465 2800 y Fo(00)537 2836 y FB(is)i(not)h(a)f Fx(\000)942 2851 y Fv(t;d)1028 2836 y FB(-form)m(ula:)40 b(The)30 b(transformation)e(from)h Fy(\015)34 b FB(to)c Fy(\014)2805 2800 y Fo(00)2877 2836 y FB(has)g(turned)h(the)257 2956 y(small)g(disjunctions)h Fx(\()p Fy(\025)1148 2971 y Fu(1)1209 2956 y Fw(_)23 b Fy(:)17 b(:)g(:)k Fw(_)i Fy(\025)1580 2971 y Fv(d)1620 2956 y Fx(\))33 b FB(on)f(the)h(b)s (ottom)e(lev)m(el)h(of)g Fy(\015)38 b FB(in)m(to)32 b(form)m(ulas)1455 3059 y Fp(^)1499 3269 y Fv(i)1583 3154 y Fy(\027)1631 3169 y Fu(1)p Fv(i)1716 3154 y Fw(_)23 b Fy(:)17 b(:)g(:)22 b Fw(_)2030 3059 y Fp(^)2074 3269 y Fv(i)2157 3154 y Fy(\027)2205 3169 y Fv(di)2270 3154 y Fy(:)257 3415 y FB(Applying)38 b(the)i(distributiv)m(e)e(la)m(w)g(to)h(all)e(these)j (subform)m(ulas)f(turns)g(them)g(in)m(to)f(big)257 3536 y(conjunctions)32 b(of)f(disjunctions)g(of)g(at)h(most)e Fy(d)i FB(literals,)d(and)j(since)g Fy(t)f FB(is)g(o)s(dd,)h(it)f (turns)257 3656 y(the)39 b(whole)g(form)m(ula)d Fy(\014)1138 3620 y Fo(00)1219 3656 y FB(in)m(to)i(a)g Fx(\000)1571 3671 y Fv(t;d)1656 3656 y FB(-form)m(ula)e Fy(\014)2113 3620 y Fo(0)2136 3656 y FB(.)62 b(Since)38 b Fy(d)h FB(is)f(\034xed,)j (the)e(size)g(only)257 3776 y(increases)30 b(p)s(olynomially)-8 b(,)26 b(and)j Fy(\014)1507 3740 y Fo(0)1559 3776 y FB(can)g(b)s(e)g (computed)g(from)f Fy(\014)2597 3740 y Fo(00)2668 3776 y FB(in)g(p)s(olynomial)d(time.)257 3897 y(And)33 b(w)m(e)h(still)c(ha) m(v)m(e:)45 b Fy(\015)37 b FB(is)32 b(satis\034able)g(if)g(and)g(only)g (if)g Fy(\014)2349 3861 y Fo(0)2404 3897 y FB(has)h(a)f(go)s(o)s(d)g (assignmen)m(t.)404 4017 y(All)j(that)h(remains)f(to)i(do)f(is)g(add)h (a)f(subform)m(ula)g(stating)f(that)i(all)d(assignmen)m(ts)257 4137 y(of)e(w)m(eigh)m(t)h Fy(k)j FB(are)c(go)s(o)s(d.)43 b(W)-8 b(e)33 b(let)1301 4405 y Fy(\013)28 b Fx(=)1532 4281 y Fv(k)1496 4310 y Fp(^)1495 4520 y Fv(i)p Fu(=1)1692 4310 y Fp(^)1625 4522 y Fv(S;T)10 b Fo(\022X)1844 4532 y Fi(i)1672 4590 y Fv(S)t Fo(6)p Fu(=)p Fv(T)1870 4405 y Fx(\()p Fw(:)p Fy(Y)2053 4364 y Fv(S)2031 4430 y(i)2126 4405 y Fw(_)22 b(:)p Fy(Y)2359 4364 y Fv(T)2337 4430 y(i)2414 4405 y Fx(\))257 4755 y FB(and)33 b Fy(\014)g Fx(=)28 b Fy(\013)23 b Fw(^)f Fy(\014)873 4719 y Fo(0)896 4755 y FB(.)44 b(Then)33 b Fy(\014)38 b FB(is)32 b(\(equiv)-5 b(alen)m(t)33 b(to\))f(a)g Fx(\000)2213 4770 y Fv(t;d)2298 4755 y FB(-form)m(ula)f(that)h(satis\034es)h(\()p FA(??)p FB(\).)404 4952 y(Next,)g(w)m(e)h(pro)m(v)m(e)f(W)p Fx([)p Fy(t)p Fx(])c Fw(\022)f FB(M)p Fx([)p Fy(t)22 b Fx(+)g(1])p FB(.)43 b(F)-8 b(or)32 b(simplicit)m(y)-8 b(,)30 b(let)h(us)j(assume)e (again)f(that)257 5073 y Fy(t)f FB(is)e(o)s(dd.)42 b(Let)30 b Fy(d)d Fx(=)h(2)h FB(if)e Fy(t)h Fx(=)g(1)h FB(and)g Fy(d)e Fx(=)h(1)h FB(otherwise.)43 b(Recall)27 b(that)i(b)m(y)h (Theorem)f FA(??)p FB(,)257 5193 y Ft(WSa)-8 b(t)o Fx(\(\000)635 5152 y Fo(\000)635 5221 y Fv(t;d)720 5193 y Fx(\))33 b FB(is)f(W)p Fx([)p Fy(t)p Fx(])p FB(-complete.)42 b(W)-8 b(e)33 b(shall)e(pro)m(v)m(e)j(that)1099 5388 y Fz(p-)p Ft(WSa)-8 b(t)n Fx(\(\000)1561 5347 y Fo(\000)1561 5416 y Fv(t;d)1646 5388 y Fx(\))28 b Fw(\024)1789 5347 y Fr(fpt)1909 5388 y Fz(p-lo)-5 b(g-)o Ft(Sa)d(t)o Fx(\(\000)2414 5403 y Fv(t)p Fu(+1)q Fv(;)p Fu(1)2589 5388 y Fx(\))p Fy(:)641 b FB(\(4.3\))p eop %%Page: 16 16 16 15 bop 257 573 a FB(W)-8 b(e)33 b(simply)e(rev)m(erse)k(the)e(idea)f (of)g(the)h(pro)s(of)f(that)g(M)p Fx([)p Fy(t)p Fx(])c Fw(\022)g FB(W)q Fx([)p Fy(t)p Fx(])p FB(.)404 693 y(Let)k Fy(\014)i Fw(2)28 b Fx(\000)822 652 y Fo(\000)822 721 y Fv(t;d)940 693 y FB(and)k Fy(k)f Fw(\025)d Fx(1)p FB(,)33 b(sa)m(y)-8 b(,)1188 942 y Fy(\014)34 b Fx(=)1410 847 y Fp(^)1380 1059 y Fv(i)1404 1068 y Fh(1)1439 1059 y Fo(2)p Fv(I)1517 1068 y Fh(1)1598 847 y Fp(_)1568 1059 y Fv(i)1592 1068 y Fh(2)1626 1059 y Fo(2)p Fv(I)1704 1068 y Fh(2)1755 942 y Fy(:)17 b(:)g(:)1910 847 y Fp(^)1886 1059 y Fv(i)1910 1067 y Fi(t)1938 1059 y Fo(2)p Fv(I)2016 1067 y Fi(t)2060 942 y Fy(\016)t Fx(\()p Fy(i)2178 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