(original) (raw)

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b(enco)s(ding)h(sc)m(heme)h(is)g(called)f(the)h(long)f(co)s(de:)41 b(for)30 b(ev)m(ery)h Fm(i)25 b Fl(2)g Fo([)p Fm(n)p Fo(],)30 b Fm(i)h Fo(is)e(enco)s(ded)g(b)m(y)h(a)h(2)3553 5124 y Fk(n)0 5270 y Fo(bit)g(string)g(whic)m(h)g(is)g(the)h(truth)f (table)h(of)h(the)f(c)m(haracter)h Fm(\037)2103 5289 y Fj(f)p Fk(i)p Fj(g)2202 5270 y Fo(.)46 b(The)31 b(long)h(co)s(de)g(w) m(as)h(in)m(tro)s(duced)d(b)m(y)0 5383 y(Bellare,)g(Goldreic)m(h)g(and) g(Sudan)e([1)q(].)1777 5656 y(1)p eop %%Page: 2 2 2 1 bop 3555 100 a Fo(2)0 511 y(The)30 b(v)m(eri\014er)f(w)m(orks)i(as) f(follo)m(ws:)136 699 y Fl(\017)46 b Fo(Pic)m(k)31 b(a)f(random)g(edge) h(\()p Fm(u;)15 b(v)s Fo(\))27 b Fl(2)e Fm(E)5 b Fo(.)136 887 y Fl(\017)46 b Fo(Let)31 b Fm(A)g Fo(b)s(e)f(the)h(table)f(\\supp)s (osed)f(to)i(b)s(e")g Fm(\037)1773 905 y Fj(f)p Fk(l)q Fh(\()p Fk(v)r Fh(\))p Fj(g)1991 887 y Fo(and)f(let)h Fm(B)k Fo(b)s(e)30 b(the)g(table)h(\\supp)s(osed)e(to)i(b)s(e")227 1000 y Fm(\037)284 1018 y Fj(f)p Fk(l)q Fh(\()p Fk(w)r 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b(Hence,)35 b(b)m(y)e(Theorem)g(1,)0 2452 y(Consistency-T)-8 b(est\()p Fm(A;)15 b(B)5 b(;)15 b(\031)982 2466 y Fk(v)r(w)1076 2452 y Fo(\))31 b(accepts)h(with)d(probabilit)m(y)f(at)j(least)f(1)21 b Fl(\000)f Fm(\017)p Fo(.)40 b Fb(2)0 2690 y Fg(Lemma)34 b(3)0 2803 y Fc(Soundness:)39 b Ff(If)26 b Fm(O)s(P)13 b(T)g Fo(\()p Fl(L)p Fo(\))25 b Fl(\024)g Fm(\016)30 b Ff(and)c Fm(\016)j Ff(is)d(su\016cien)m(tly)e(small)h(\(as)i(a)f (function)f(of)h Fm(\017;)15 b(\021)s Ff(\),)28 b(then)e(for)g(ev)m (ery)0 2916 y(pro)s(of)k Fo(\005)p Ff(,)g(Pr[)p Fo(\005)h Ff(is)e(accepted])p Fl(\024)1149 2880 y Fh(1)p 1149 2895 36 4 v 1149 2947 a(2)1214 2916 y Fo(+)20 b(2)p Fm(\021)s Ff(.)0 3128 y Fg(Pr)n(oof:)37 b Fo(W)-8 b(e)26 b(pro)m(v)m(e)f(the)g (con)m(trap)s(ositiv)m(e,)h(that)f(if)e(on)i(an)f(instance)g Fl(L)g Fo(of)h(lab)s(el)e(co)m(v)m(er,)28 b(there)c(is)g(a)h(pro)s(of)0 3241 y(\005)j(suc)m(h)f(that)h(the)g(v)m(eri\014er)f(accepts)i(\005)f (with)e(probabilit)m(y)f(more)j(than)2470 3205 y Fh(1)p 2470 3220 V 2470 3273 a(2)2530 3241 y Fo(+)15 b(2)p Fm(\021)s Fo(,)28 b(then)g Fl(L)f Fo(has)g(a)h(lab)s(eling)0 3354 y(that)35 b(satis\014es)f(\012\()p Fm(\017)678 3321 y Fh(2)733 3354 y Fo(log)850 3317 y Fj(\000)p Fh(2)945 3354 y Fo(\()991 3318 y Fh(1)p 990 3333 38 4 v 990 3385 a Fk(\021)1038 3354 y Fo(\))p Fm(\021)1121 3321 y Fh(3)1161 3354 y Fo(\))h(fraction)f(of)h(its)f(edges.)54 b(This)33 b(giv)m(es)i(a)g(con)m(tradiction)g(pro)m(vided)e Fm(\016)0 3467 y Fo(w)m(as)e(c)m(hosen)g(su\016cien)m(tly)e(small.)0 3693 y(Supp)s(ose)38 b(Pr[\005)i(is)f(accepted])p Fl(\025)1189 3657 y Fh(1)p 1189 3672 36 4 v 1189 3724 a(2)1261 3693 y Fo(+)26 b(2)p Fm(\021)s Fo(,)43 b(then)d(for)g(at)h(least)f Fm(\021)j Fo(fraction)d(of)g(the)g(edges,)j(the)e(prob-)0 3806 y(abilit)m(y)33 b(that)j(Consistency-T)-8 b(est)35 b(accepts)h(the)g(tables)e(for)h(that)h(edge)f(is)f(at)i(least)2945 3770 y Fh(1)p 2945 3785 V 2945 3837 a(2)3014 3806 y Fo(+)23 b Fm(\021)38 b Fo(\(call)c(these)0 3919 y(edges)d(\\go)s(o)s(d"\).)42 b(By)30 b(Theorem)h(1,)g(this)e(means)h(that)h(for)f(ev)m(ery)h(go)s(o) s(d)g(edge)g(\()p Fm(v)s(;)15 b(w)r Fo(\):)1505 4050 y Fe(X)1361 4251 y Fk(\013)p Fj(\022)p Fh([)p Fk(n)p Fh(])p Fk(;\014)s Fj(\022)p Fh([)p Fk(m)p Fh(])p Fk(;)1250 4371 y Fj(j)p Fk(\013)p Fj(j)p Fk(;)p Fj(j)p Fk(\014)s Fj(j\024)p Fk(O)r Fh(\()1586 4335 y(1)p 1586 4350 V 1589 4402 a Fk(\017)1641 4371 y Fh(log)q(\()1771 4335 y(1)p 1769 4350 38 4 v 1769 4402 a Fk(\021)1818 4371 y Fh(\)\))p Fk(;)1451 4489 y(\013)p Fj(\022)p Fk(\031)r Fh(\()p Fk(\014)s Fh(\))1931 4113 y Fo(^)1907 4136 y Fm(A)1975 4099 y Fh(2)1975 4159 y Fk(\013)2046 4113 y Fo(^)2025 4136 y Fm(B)2099 4099 y Fh(2)2094 4160 y Fk(\014)2166 4136 y Fl(\025)25 b Fm(\021)2310 4099 y Fh(2)3484 4136 y Fo(\(1\))0 4683 y(where)j Fm(A)h Fo(and)f Fm(B)34 b Fo(are)29 b(the)g(tables)f(for)h Fm(l)r Fo(\()p Fm(v)s Fo(\))g(and)g Fm(l)r Fo(\()p Fm(w)r Fo(\))g(resp)s(ectiv)m(ely)-8 b(.)40 b(Using)28 b(this)g(w)m(e)h(giv)m (es)g(a)g(random-)0 4796 y(ized)37 b(lab)s(eling)f(for)h Fl(L)h Fo(that)g(satis\014es)g(a)g(signi\014can)m(t)f(fraction)g(of)h (edges)h(in)d(exp)s(ectation.)64 b(It)38 b(follo)m(ws)0 4908 y(that)31 b(there)g(m)m(ust)f(exist)g(a)h(lab)s(eling)d(that)j (satis\014es)e(at)j(least)e(that)h(m)m(uc)m(h)g(fraction)f(of)g(edges.) 0 5134 y(F)-8 b(or)31 b Fm(v)e Fl(2)c Fm(V)20 b Fo(,)30 b Fm(l)r Fo(\()p Fm(v)s Fo(\))i(is)d(de\014ned)g(as)i(follo)m(ws:)141 5360 y(Lo)s(ok)g(at)g(T)-8 b(able)30 b Fm(A)p Fo(.)p eop %%Page: 3 3 3 2 bop 3555 100 a Fo(3)141 399 y(Pic)m(k)34 b Fm(\013)e Fl(\022)g Fo([)p Fm(n)p Fo(])i(with)f(probabilit)m(y)1383 376 y(^)1359 399 y Fm(A)1427 366 y Fh(2)1427 421 y Fk(\013)1511 399 y Fo(\(note)i(that)1953 330 y Fe(P)2049 425 y Fk(\013)p Fj(\022)p Fh([)p Fk(n)p Fh(])2275 376 y Fo(^)2251 399 y Fm(A)2319 366 y Fh(2)2319 421 y Fk(\013)2400 399 y Fo(=)c(1)k(and)e(therefore)i(the)f(squares)0 511 y(of)d(F)-8 b(ourier)29 b(co)s(e\016cien)m(ts)j(de\014ne)d(a)i(probabilit)m(y)d (distribution\).)141 624 y(If)i Fl(j)p Fm(\013)p Fl(j)c(\025)f Fm(O)s Fo(\()579 589 y Fh(1)p 579 604 36 4 v 582 656 a Fk(\017)640 624 y Fo(log)q(\()803 589 y Fh(1)p 802 604 38 4 v 802 656 a Fk(\021)850 624 y Fo(\)\))31 b(then)f(giv)m(e)h (up.)141 737 y(Else,)f(de\014ne)g Fm(l)r Fo(\()p Fm(v)s Fo(\))h(to)g(b)s(e)f(a)h(random)e(elemen)m(t)i(of)g(the)f(set)h Fm(\013)p Fo(.)0 963 y(F)-8 b(or)31 b Fm(w)d Fl(2)d Fm(W)13 b Fo(,)30 b Fm(l)r Fo(\()p Fm(w)r Fo(\))h(is)f(de\014ned)f(as)h(follo)m (ws:)141 1189 y(Lo)s(ok)h(at)g(T)-8 b(able)30 b Fm(B)5 b Fo(.)141 1302 y(Pic)m(k)30 b Fm(\014)h Fl(\022)25 b Fo([)p Fm(m)p Fo(])30 b(w.p.)911 1279 y(^)890 1302 y Fm(B)964 1269 y Fh(2)959 1329 y Fk(\014)1006 1302 y Fo(.)141 1429 y(If)g Fl(j)p Fm(\014)5 b Fl(j)26 b(\025)f Fm(O)s Fo(\()577 1393 y Fh(1)p 577 1408 36 4 v 580 1461 a Fk(\017)638 1429 y Fo(log)q(\()801 1393 y Fh(1)p 800 1408 38 4 v 800 1461 a Fk(\021)848 1429 y Fo(\)\))31 b(then)f(giv)m(e)h(up.)141 1542 y(Else,)f(de\014ne)g Fm(l)r Fo(\()p Fm(w)r Fo(\))h(to)g(b)s(e)f(a) h(random)e(elemen)m(t)i(of)g(the)f(set)h Fm(\014)5 b Fo(.)0 1768 y(Note)45 b(that)e(w)m(e)h(are)g(assuming)e(that)i Fm(\013;)15 b(\014)52 b Fl(6)p Fo(=)47 b Fm(\036)p Fo(.)79 b(W)-8 b(e)45 b(will)40 b(come)45 b(bac)m(k)f(to)g(this)e(in)g(the)h (pro)s(of)g(of)0 1881 y(the)31 b(consistency)f(test.)0 1994 y(No)m(w,)g(from)f(\(1\))h(w)m(e)g(see)f(that)h(for)f(a)h(go)s(o)s (d)f(edge)g(\()p Fm(v)s(;)15 b(w)r Fo(\),)32 b(with)c(probabilit)m(y)e (at)k(least)g Fm(\021)3023 1961 y Fh(2)3092 1994 y Fo(w)m(e)f(pic)m(k)g (for)g Fm(v)0 2107 y Fo(a)i(set)g Fm(\013)f Fo(and)g(for)g Fm(w)j Fo(a)e(set)g Fm(\014)k Fo(s.t.)42 b Fl(j)p Fm(\013)p Fl(j)p Fm(;)15 b Fl(j)p Fm(\014)5 b Fl(j)27 b Fm(<)e(O)s Fo(\()1681 2071 y Fh(1)p 1681 2086 36 4 v 1684 2138 a Fk(\017)1741 2107 y Fo(log)r(\()1905 2071 y Fh(1)p 1904 2086 38 4 v 1904 2138 a Fk(\021)1952 2107 y Fo(\)\))31 b(and)f(s.t.)41 b Fm(\013)26 b Fl(\022)f Fm(\031)s Fo(\()p Fm(\014)5 b Fo(\).)0 2239 y(In)33 b(particular,)g Fl(9)h Fm(i)682 2253 y Fh(0)753 2239 y Fl(2)c Fm(\013;)15 b(j)979 2253 y Fh(0)1051 2239 y Fl(2)31 b Fm(\014)39 b Fo(s.t.)51 b Fm(\031)s Fo(\()p Fm(j)1532 2253 y Fh(0)1573 2239 y Fo(\))31 b(=)g Fm(i)1772 2253 y Fh(0)1812 2239 y Fo(.)51 b(With)33 b(probabilit)m(y)f(\012\()p Fm(\017)2733 2206 y Fh(2)2787 2239 y Fo(log)2905 2202 y Fj(\000)p Fh(2)2999 2239 y Fo(\()3045 2203 y Fh(1)p 3044 2218 V 3044 2270 a Fk(\021)3092 2239 y Fo(\)\))i(w)m(e)h(pic)m(k)e Fm(i)3560 2253 y Fh(0)0 2352 y Fo(to)g(b)s(e)e Fm(l)r Fo(\()p Fm(v)s Fo(\))i(and)e Fm(j)632 2366 y Fh(0)704 2352 y Fo(to)i(b)s(e)e Fm(l)r Fo(\()p Fm(w)r Fo(\),)j(and)d(then)h(the)g(edge)h(\()p Fm(v)s(;)15 b(w)r Fo(\))34 b(is)d(satis\014ed.)45 b(Recall)31 b(that)i(the)f(fraction)0 2465 y(of)h(go)s(o)s(d)g(edges)g(is)f(at)i (least)f Fm(\021)s Fo(,)h(and)f(for)f(eac)m(h)j(go)s(o)s(d)d(edge)i(w)m (e)f(select)h(a)f(go)s(o)s(d)g(pair)f(of)h(sets)g Fm(\013;)15 b(\014)39 b Fo(with)0 2578 y(probabilit)m(y)29 b(at)j(least)g Fm(\021)840 2545 y Fh(2)880 2578 y Fo(.)44 b(Hence)32 b(it)f(follo)m(ws)g(that)h(the)g(fraction)f(of)h(edges)g(satis\014ed)e (b)m(y)i(the)f(lab)s(eling)0 2691 y(describ)s(ed)d(ab)s(o)m(v)m(e)k(is) d(at)i(least)g(\012\()p Fm(\017)1213 2658 y Fh(2)1267 2691 y Fo(log)1385 2654 y Fj(\000)p Fh(2)1479 2691 y Fo(\()1525 2655 y Fh(1)p 1524 2670 V 1524 2722 a Fk(\021)1572 2691 y Fo(\))p Fm(\021)1655 2658 y Fh(3)1696 2691 y Fo(\).)41 b Fb(2)0 2966 y Fo(It)30 b(remains)g(to)h(pro)m(v)m(e)g(Theorem)f(1.)0 3192 y Fg(Pr)n(oof:)40 b Fo(\(of)31 b(Theorem)f(1\))0 3305 y(F)-8 b(or)31 b Fm(x)25 b Fl(2)g(f\006)p Fo(1)p Fl(g)532 3272 y Fk(n)580 3305 y Fo(,)31 b(de\014ne)e Fm(x)21 b Fl(\016)f Fm(\031)29 b Fl(2)24 b(f\006)p Fo(1)p Fl(g)1408 3272 y Fk(m)1506 3305 y Fo(to)31 b(b)s(e)f(the)h(string)e (s.t.)41 b(\()p Fm(x)21 b Fl(\016)f Fm(\031)s Fo(\))2581 3319 y Fk(j)2644 3305 y Fo(=)25 b Fm(x)2792 3323 y Fk(\031)r Fh(\()p Fk(j)t Fh(\))2926 3305 y Fo(,)30 b(for)h Fm(j)f Fo(=)25 b(1)p Fm(:::m)p Fo(.)0 3531 y Fc(T)-9 b(est:)141 3644 y Fo(Pic)m(k)30 b(at)i(random)d Fm(x)c Fl(2)g(f\006)p Fo(1)p Fl(g)1158 3611 y Fk(n)1206 3644 y Fo(,)31 b Fm(y)d Fl(2)d(f\006)p Fo(1)p Fl(g)1627 3611 y Fk(m)141 3757 y Fo(Pic)m(k)30 b Fm(\026)25 b Fl(2)g(f\006)p Fo(1)p Fl(g)716 3724 y Fk(m)784 3757 y Fo(,)31 b(s.t.)41 b Fl(8)p Fm(j)30 b Fo(=)25 b(1)p Fm(:::m)p Fo(,)1335 4038 y Fm(\026)1390 4052 y Fk(j)1452 4038 y Fo(=)1548 3883 y Fe(\()1621 3975 y Fo(+1)121 b Fn(w.p.)41 b Fo(1)21 b Fl(\000)f Fm(\017)1621 4110 y Fl(\000)p Fo(1)121 b Fn(w.p.)41 b Fm(\017)141 4319 y Fo(Let)31 b Fm(z)f Fo(=)25 b Fm(y)s Fo(\()p Fm(x)20 b Fl(\016)h Fm(\031)s Fo(\))p Fm(\026)141 4432 y Fo(Accept)32 b(i\013)d Fm(B)5 b Fo(\()p Fm(z)t Fo(\))26 b(=)f Fm(A)p Fo(\()p Fm(x)p Fo(\))p Fm(B)5 b Fo(\()p Fm(y)s Fo(\))0 4658 y Fc(Completeness:)39 b Fo(Assume)30 b Fm(A)c Fo(=)f Fm(\037)1280 4677 y Fj(f)p Fk(i)1339 4686 y Fi(0)1373 4677 y Fj(g)1413 4658 y Fm(;)15 b(B)30 b Fo(=)25 b Fm(\037)1705 4677 y Fj(f)p Fk(j)1769 4686 y Fi(0)1803 4677 y Fj(g)1843 4658 y Fm(;)15 b(\031)s Fo(\()p Fm(j)2010 4672 y Fh(0)2050 4658 y Fo(\))26 b(=)f Fm(i)2238 4672 y Fh(0)2278 4658 y Fo(.)40 b(Then)141 4771 y Fm(A)p Fo(\()p Fm(x)p Fo(\))26 b(=)f Fm(x)505 4785 y Fk(i)529 4794 y Fi(0)568 4771 y Fo(,)141 4884 y Fm(B)5 b Fo(\()p Fm(y)s Fo(\))25 b(=)g Fm(y)499 4898 y Fk(j)528 4907 y Fi(0)566 4884 y Fo(,)31 b(and)141 4997 y Fm(B)5 b Fo(\()p Fm(z)t Fo(\))26 b(=)f Fm(z)495 5011 y Fk(j)524 5020 y Fi(0)588 4997 y Fo(=)g Fm(y)729 5011 y Fk(j)758 5020 y Fi(0)795 4997 y Fo(\()p Fm(x)c Fl(\016)g Fm(\031)s Fo(\))1059 5011 y Fk(j)1088 5020 y Fi(0)1126 4997 y Fm(\026)1181 5011 y Fk(j)1210 5020 y Fi(0)1248 4997 y Fo(.)0 5110 y(But)30 b(\()p Fm(x)19 b Fl(\016)h Fm(\031)s Fo(\))441 5124 y Fk(j)470 5133 y Fi(0)534 5110 y Fo(=)k Fm(x)681 5128 y Fk(\031)r Fh(\()p Fk(j)780 5137 y Fi(0)815 5128 y Fh(\))872 5110 y Fo(=)h Fm(x)1020 5124 y Fk(i)1044 5133 y Fi(0)1082 5110 y Fo(,)30 b(so)g(the)g(test)h(reduces)e(to)i(accepting)f(i\013)f Fm(\026)2581 5124 y Fk(j)2610 5133 y Fi(0)2674 5110 y Fo(=)24 b(+1,)31 b(whic)m(h)d(holds)g Fn(w.p.)0 5223 y Fo(1)21 b Fl(\000)f Fm(\017)p Fo(.)p eop %%Page: 4 4 4 3 bop 3555 100 a Fo(4)0 399 y Fc(Soundness:)39 b Fo(Note)27 b(the)f(follo)m(wing)e(expression)h(for)g(the)h(probabilit)m(y)d(that)k (the)f(test)g(accepts)h Fm(A;)15 b(B)5 b(;)15 b(\031)s Fo(:)402 645 y(Pr[accept])27 b(=)967 653 y Fm(E)921 708 y Fk(x;y)r(;\026)1082 645 y Fo([)1117 584 y(1)21 b(+)f Fm(A)p Fo(\()p Fm(x)p Fo(\))p Fm(B)5 b Fo(\()p Fm(y)s Fo(\))p Fm(B)g Fo(\()p Fm(y)s Fo(\()p Fm(x)21 b Fl(\016)g Fm(\031)s Fo(\))p Fm(\026)p Fo(\))p 1117 625 1050 4 v 1620 708 a(2)2177 645 y(])828 884 y(=)934 822 y(1)p 934 863 46 4 v 934 946 a(2)1009 884 y(+)1110 822 y(1)p 1110 863 V 1110 946 a(2)1224 892 y Fm(E)1178 947 y Fk(x;y)r(;\026)1339 884 y Fo([)1429 797 y Fe(X)1403 999 y Fk(\013)p Fj(\022)p Fh([)p Fk(n)p Fh(])1364 1089 y Fk(\014)s(;\015)t Fj(\022)p Fh([)p Fk(m)p Fh(])1663 861 y Fo(^)1639 884 y Fm(A)1707 898 y Fk(\013)1778 861 y Fo(^)1757 884 y Fm(B)1826 899 y Fk(\014)1894 861 y Fo(^)1873 884 y Fm(B)1942 898 y Fk(\015)1986 884 y Fm(\037)2043 898 y Fk(\013)2093 884 y Fo(\()p Fm(x)p Fo(\))p Fm(\037)2272 899 y Fk(\014)2319 884 y Fo(\()p Fm(y)s Fo(\))p Fm(\037)2494 898 y Fk(\015)2539 884 y Fo(\()p Fm(y)s Fo(\()p Fm(x)g Fl(\016)f Fm(\031)s Fo(\))p Fm(\026)p Fo(\)])828 1268 y(=)934 1207 y(1)p 934 1247 V 934 1331 a(2)1009 1268 y(+)1110 1207 y(1)p 1110 1247 V 1110 1331 a(2)1199 1182 y Fe(X)1181 1380 y Fk(\013;\014)s(;\015)1388 1245 y Fo(^)1364 1268 y Fm(A)1432 1282 y Fk(\013)1503 1245 y Fo(^)1481 1268 y Fm(B)1550 1283 y Fk(\014)1619 1245 y Fo(^)1597 1268 y Fm(B)1666 1282 y Fk(\015)1769 1276 y Fm(E)1723 1331 y Fk(x;y)r(;\026)1885 1268 y Fo([)p Fm(\037)1967 1282 y Fk(\013)2016 1268 y Fo(\()p Fm(x)p Fo(\))p Fm(\037)2195 1283 y Fk(\014)2243 1268 y Fo(\()p Fm(y)s Fo(\))p Fm(\037)2418 1282 y Fk(\015)2463 1268 y Fo(\()p Fm(y)s Fo(\))p Fm(\037)2638 1282 y Fk(\015)2682 1268 y Fo(\()p Fm(x)h Fl(\016)g Fm(\031)s Fo(\))p Fm(\037)3003 1282 y Fk(\015)3047 1268 y Fo(\()p Fm(\026)p Fo(\)])3484 1514 y(\(2\))0 1831 y(W)-8 b(e)32 b(need)e(the)g(follo)m(wing)f(lemma)h (to)h(pro)s(ceed:)0 2019 y Fg(Lemma)j(4)0 2131 y Fm(\037)57 2145 y Fk(\015)101 2131 y Fo(\()p Fm(x)7 b Fl(\016)g Fm(\031)s Fo(\))26 b(=)f Fm(\037)516 2150 y Fk(\031)557 2159 y Fi(2)591 2150 y Fh(\()p Fk(\015)t Fh(\))691 2131 y Fo(\()p Fm(x)p Fo(\))p Ff(,)h(where)d Fm(\031)1172 2145 y Fh(2)1211 2131 y Fo(\()p Fm(\015)5 b Fo(\))26 b Fl(\022)f Fo([)p Fm(n)p Fo(])f Ff(is)e(de\014ned)h(as)h Fm(\031)2132 2145 y Fh(2)2171 2131 y Fo(\()p Fm(\015)5 b Fo(\))26 b(=)f Fl(f)p Fm(i)h Fo(:)g Fm(i)f Fl(2)g Fo([)p Fm(n)p Fo(])p Fm(;)15 b Fl(j)p Fm(\031)2935 2099 y Fj(\000)p Fh(1)3030 2131 y Fo(\()p Fm(i)p Fo(\))7 b Fl(\\)g Fm(\015)e Fl(j)31 b Ff(is)f(o)s(dd)o Fl(g)0 2319 y Fg(Pr)n(oof:)902 2432 y Fm(\037)959 2446 y Fk(\015)1004 2432 y Fo(\()p Fm(x)20 b Fl(\016)h Fm(\031)s Fo(\))k(=)1388 2346 y Fe(Y)1392 2543 y Fk(l)q Fj(2)p Fk(\015)1505 2432 y Fo(\()p Fm(x)20 b Fl(\016)h Fm(\031)s Fo(\))1768 2447 y Fk(l)1819 2432 y Fo(=)1915 2346 y Fe(Y)1919 2543 y Fk(l)q Fj(2)p Fk(\015)2047 2432 y Fm(x)2099 2451 y Fk(\031)r Fh(\()p Fk(l)q Fh(\))2248 2432 y Fo(=)k Fm(\037)2401 2451 y Fk(\031)2442 2460 y Fi(2)2476 2451 y Fh(\()p Fk(\015)t Fh(\))2575 2432 y Fo(\()p Fm(x)p Fo(\))0 2668 y Fb(2)0 2831 y Fo(Hence)526 3050 y(Pr[accept])i(=)1058 2989 y(1)p 1058 3030 V 1058 3113 a(2)1134 3050 y(+)1234 2989 y(1)p 1234 3030 V 1234 3113 a(2)1323 2964 y Fe(X)1305 3162 y Fk(\013;\014)s(;\015)1512 3027 y Fo(^)1488 3050 y Fm(A)1556 3064 y Fk(\013)1627 3027 y Fo(^)1606 3050 y Fm(B)1675 3065 y Fk(\014)1743 3027 y Fo(^)1722 3050 y Fm(B)1791 3064 y Fk(\015)1893 3058 y Fm(E)1848 3113 y Fk(x;y)r(;\026)2009 3050 y Fo([)p Fm(\037)2091 3065 y Fk(\014)s Fj(4)p Fk(\015)2241 3050 y Fo(\()p Fm(y)s Fo(\))p Fm(\037)2416 3069 y Fk(\013)p Fj(4)p Fk(\031)2566 3078 y Fi(2)2600 3069 y Fh(\()p Fk(\015)t Fh(\))2699 3050 y Fo(\()p Fm(x)p Fo(\))p Fm(\037)2878 3064 y Fk(\015)2923 3050 y Fo(\()p Fm(\026)p Fo(\)])0 3391 y(The)j(sum)f(is)h(zero)h(unless)e Fm(\014)h Fo(=)25 b Fm(\015)5 b(;)15 b(\013)26 b Fo(=)f Fm(\031)1433 3405 y Fh(2)1472 3391 y Fo(\()p Fm(\015)5 b Fo(\).)42 b(Also,)30 b(w)m(e)h(ha)m(v)m(e)h Fm(E)2302 3405 y Fk(\026)2348 3391 y Fo([)p Fm(\037)2430 3405 y Fk(\015)2475 3391 y Fo(\()p Fm(\026)p Fo(\)])26 b(=)f(\(1)c Fl(\000)f Fo(2)p Fm(\017)p Fo(\))3056 3358 y Fj(j)p Fk(\015)t Fj(j)3140 3391 y Fo(.)41 b(Hence,)526 3807 y(Pr[accept])27 b(=)1058 3746 y(1)p 1058 3786 V 1058 3870 a(2)1134 3807 y(+)1234 3746 y(1)p 1234 3786 V 1234 3870 a(2)1376 3721 y Fe(X)1388 3919 y Fk(\013;\014)1305 4006 y(\013)p Fh(=)p Fk(\031)1446 4015 y Fi(2)1480 4006 y Fh(\()p Fk(\014)s Fh(\))1617 3784 y Fo(^)1593 3807 y Fm(A)1661 3821 y Fk(\013)1733 3784 y Fo(^)1711 3807 y Fm(B)1785 3770 y Fh(2)1780 3830 y Fk(\014)1827 3807 y Fo(\(1)21 b Fl(\000)f Fo(2)p Fm(\017)p Fo(\))2136 3770 y Fj(j)p Fk(\014)s Fj(j)0 4234 y Fo(If)30 b(the)g(probabilit)m(y)e(of)j(acceptance)i(is)c(at)i(least)1701 4199 y Fh(1)p 1701 4214 36 4 v 1701 4266 a(2)1766 4234 y Fo(+)20 b Fm(\021)s Fo(,)31 b(w)m(e)g(get:)712 4589 y(1)p 712 4630 46 4 v 712 4713 a(2)787 4651 y(+)20 b Fm(\021)29 b Fl(\024)1058 4589 y Fo(1)p 1058 4630 V 1058 4713 a(2)1134 4651 y(+)1234 4589 y(1)p 1234 4630 V 1234 4713 a(2)1376 4564 y Fe(X)1388 4762 y Fk(\013;\014)1305 4849 y(\013)p Fh(=)p Fk(\031)1446 4858 y Fi(2)1480 4849 y Fh(\()p Fk(\014)s Fh(\))1617 4628 y Fo(^)1593 4651 y Fm(A)1661 4665 y Fk(\013)1733 4628 y Fo(^)1711 4651 y Fm(B)1785 4613 y Fh(2)1780 4674 y Fk(\014)1827 4651 y Fo(\(1)21 b Fl(\000)f Fo(2)p Fm(\017)p Fo(\))2136 4613 y Fj(j)p Fk(\014)s Fj(j)717 5003 y Fl(\))25 b Fo(2)p Fm(\021)k Fl(\024)1119 4917 y Fe(X)1131 5115 y Fk(\013;\014)1048 5202 y(\013)p Fh(=)p Fk(\031)1189 5211 y Fi(2)1223 5202 y Fh(\()p Fk(\014)s Fh(\))1360 4980 y Fo(^)1336 5003 y Fm(A)1404 5017 y Fk(\013)1476 4980 y Fo(^)1454 5003 y Fm(B)1528 4966 y Fh(2)1523 5026 y Fk(\014)1570 5003 y Fo(\(1)21 b Fl(\000)f Fo(2)p Fm(\017)p Fo(\))1879 4966 y Fj(j)p Fk(\014)s Fj(j)p eop %%Page: 5 5 5 4 bop 3555 100 a Fo(5)0 511 y(W)-8 b(e)32 b(no)m(w)e(use)g(the)h (Cauc)m(h)m(y-Sw)m(artz)g(inequalit)m(y)e(to)i(get)g(the)g(follo)m (wing:)482 691 y(4)p Fm(\021)575 654 y Fh(2)641 691 y Fl(\024)25 b Fo(\()843 605 y Fe(X)855 803 y Fk(\013;\014)772 890 y(\013)p Fh(=)p Fk(\031)913 899 y Fi(2)948 890 y Fh(\()p Fk(\014)s Fh(\))1085 669 y Fo(^)1061 691 y Fm(A)1129 705 y Fk(\013)1200 669 y Fo(^)1178 691 y Fm(B)1252 654 y Fh(2)1247 715 y Fk(\014)1294 691 y Fo(\(1)c Fl(\000)f Fo(2)p Fm(\017)p Fo(\))1603 654 y Fj(j)p Fk(\014)s Fj(j)1690 691 y Fo(\))1725 654 y Fh(2)1790 691 y Fl(\024)1957 605 y Fe(X)1969 803 y Fk(\013;\014)1886 890 y(\013)p Fh(=)p Fk(\031)2027 899 y Fi(2)2062 890 y Fh(\()p Fk(\014)s Fh(\))2199 669 y Fo(^)2175 691 y Fm(A)2243 654 y Fh(2)2243 714 y Fk(\013)2314 669 y Fo(^)2292 691 y Fm(B)2366 654 y Fh(2)2361 715 y Fk(\014)2409 691 y Fo(\(1)h Fl(\000)e Fo(2)p Fm(\017)p Fo(\))2717 654 y Fh(2)p Fj(j)p Fk(\014)s Fj(j)2855 605 y Fe(X)2899 803 y Fk(\014)3023 669 y Fo(^)3001 691 y Fm(B)3075 654 y Fh(2)3070 715 y Fk(\014)0 1068 y Fo(But)31 b(w)m(e)f(ha)m(v)m(e)524 1000 y Fe(P)620 1095 y Fk(\014)704 1045 y Fo(^)682 1068 y Fm(B)756 1035 y Fh(2)751 1096 y Fk(\014)824 1068 y Fo(=)25 b(1,)31 b(hence)1267 1185 y Fe(X)1279 1383 y Fk(\013;\014)1196 1470 y(\013)p Fh(=)p Fk(\031)1337 1479 y Fi(2)1371 1470 y Fh(\()p Fk(\014)s Fh(\))1508 1248 y Fo(^)1484 1271 y Fm(A)1552 1234 y Fh(2)1552 1294 y Fk(\013)1624 1248 y Fo(^)1602 1271 y Fm(B)1676 1234 y Fh(2)1671 1295 y Fk(\014)1718 1271 y Fo(\(1)21 b Fl(\000)f Fo(2)p Fm(\017)p Fo(\))2027 1234 y Fh(2)p Fj(j)p Fk(\014)s Fj(j)2175 1271 y Fl(\025)25 b Fo(4)p Fm(\021)2364 1234 y Fh(2)0 1695 y Fo(No)m(w,)e(for)d Fl(j)p Fm(\014)5 b Fl(j)26 b Fm(>)f(O)s Fo(\()698 1659 y Fh(1)p 698 1674 36 4 v 701 1726 a Fk(\017)759 1695 y Fo(log)q(\()922 1659 y Fh(1)p 921 1674 38 4 v 921 1726 a Fk(\021)969 1695 y Fo(\)\),)e(w)m(e)e(get)h(that)f(the)f (\\damping)f(factor")j(\(1)p Fl(\000)p Fo(2)p Fm(\017)p Fo(\))2659 1662 y Fh(2)p Fj(j)p Fk(\014)s Fj(j)2807 1695 y Fo(=)j(\(1)p Fl(\000)p Fo(2)p Fm(\017)p Fo(\))3171 1641 y Fk(O)r Fh(\()3264 1605 y(1)p 3264 1620 36 4 v 3267 1672 a Fk(\017)3322 1641 y Fh(log)q(\()3452 1605 y(1)p 3450 1620 38 4 v 3450 1672 a Fk(\021)3498 1641 y Fh(\)\))3583 1695 y Fl(\024)0 1826 y Fm(\021)48 1793 y Fh(2)88 1826 y Fo(.)40 b(Hence)29 b(w)m(e)f(ma)m(y)h(constrain)e(the) h(sum)f(further)g(b)m(y)h Fl(j)p Fm(\014)5 b Fl(j)26 b(\024)f Fm(O)s Fo(\()2256 1790 y Fh(1)p 2256 1805 36 4 v 2259 1857 a Fk(\017)2317 1826 y Fo(log)q(\()2480 1790 y Fh(1)p 2479 1805 38 4 v 2479 1857 a Fk(\021)2527 1826 y Fo(\)\),)30 b(and)d(still)f(b)s(e)h(at)i(least)f(3)p Fm(\021)3534 1793 y Fh(2)3575 1826 y Fo(.)0 1939 y(Finally)-8 b(,)29 b(w)m(e)h(ma)m(y)g(replace)g Fm(\013)c Fo(=)f Fm(\031)1196 1953 y Fh(2)1235 1939 y Fo(\()p Fm(\014)5 b Fo(\))31 b(b)m(y)f Fm(\013)25 b Fl(\022)g Fm(\031)s Fo(\()p Fm(\014)5 b Fo(\))31 b(and)e(only)g(increase)h(the)g(sum.)40 b(This)28 b(yields)g(the)0 2052 y(soundness)h(condition)g(as)h (required.)0 2278 y Fc(Remark:)39 b Fo(In)28 b(the)i(pro)s(of)e(so)h (far,)g(w)m(e)g(assumed)g(that)1923 2255 y(^)1899 2278 y Fm(A)1967 2293 y Fk(\036)2013 2278 y Fm(;)2075 2255 y Fo(^)2053 2278 y Fm(B)2122 2293 y Fk(\036)2194 2278 y Fo(=)c(0.)41 b(This)27 b(is)h(not)h(a)g(v)-5 b(alid)27 b(assumption)0 2391 y(-)37 b(indeed)e(the)j(pro)s(of)e(in)f(whic)m(h)h Fm(A)h Fo(and)f Fm(B)41 b Fo(are)d(all)d(+1s)i(is)f(alw)m(a)m(ys)i (accepted.)61 b(Hence)38 b(one)f(needs)g(a)0 2504 y(further)29 b(mec)m(hanism)h(to)h(ensure)e(that)1389 2481 y(^)1365 2504 y Fm(A)1433 2519 y Fk(\036)1480 2504 y Fm(;)1542 2481 y Fo(^)1520 2504 y Fm(B)1589 2519 y Fk(\036)1660 2504 y Fo(=)c(0.)41 b(W)-8 b(e)32 b(use)e(the)h(follo)m(wing)e(fact:)0 2664 y Fg(Lemma)34 b(5)0 2777 y Ff(If)42 b(table)g Fm(A)g Ff(is)f(suc)m(h)h(that)h(for)f(all)f Fm(x)p Ff(,)k Fm(A)p Fo(\()p Fl(\000)p Fm(x)p Fo(\))g(=)g Fl(\000)p Fm(A)p Fo(\()p Fm(x)p Fo(\))p Ff(,)h(then)2389 2754 y Fo(^)2365 2777 y Fm(A)2433 2792 y Fk(\036)2524 2777 y Fo(=)e(0)p Ff(.)77 b(F)-8 b(urthermore,)45 b(for)d(all)0 2890 y Fm(i)26 b Fo(=)f(1)p Fm(;)15 b(::;)g(n)p Ff(,)31 b(the)g(c)m(haracter)h Fm(\037)1050 2908 y Fj(f)p Fk(i)p Fj(g)1179 2890 y Ff(satis\014es)e Fm(\037)p Fo(\()p Fl(\000)p Fm(x)p Fo(\))25 b(=)g Fl(\000)p Fm(\037)p Fo(\()p Fm(x)p Fo(\))31 b Ff(for)f(all)f Fm(x)p Ff(.)0 3050 y Fg(Pr)n(oof:)40 b Fo(If)30 b Fm(A)p Fo(\()p Fl(\000)p Fm(x)p Fo(\))c(=)f Fl(\000)p Fm(A)p Fo(\()p Fm(x)p Fo(\))30 b(for)h(all)e Fm(x)p Fo(,)h(then)483 3199 y(^)459 3222 y Fm(A)527 3237 y Fk(\036)599 3222 y Fo(=)25 b Fm(E)762 3236 y Fk(x)806 3222 y Fo([)p Fm(A)p Fo(\()p Fm(x)p Fo(\))p Fm(\037)1078 3237 y Fk(\036)1125 3222 y Fo(\()p Fm(x)p Fo(\)])h(=)f Fm(E)1461 3236 y Fk(x)1505 3222 y Fo([)p Fm(A)p Fo(\()p Fm(x)p Fo(\)])h(=)f Fm(E)1934 3236 y Fk(x)1978 3222 y Fo([)p Fm(A)p Fo(\()p Fl(\000)p Fm(x)p Fo(\)])i(=)e Fm(E)2479 3236 y Fk(x)2523 3222 y Fo([)p Fl(\000)p Fm(A)p Fo(\()p Fm(x)p Fo(\)])h(=)f Fl(\000)3051 3199 y Fo(^)3027 3222 y Fm(A)3095 3237 y Fk(\036)0 3393 y Fo(Also,)30 b(for)g(an)m(y)h Fm(i)p Fo(,)g Fm(\037)685 3412 y Fj(f)p Fk(i)p Fj(g)784 3393 y Fo(\()p Fl(\000)p Fm(x)p Fo(\))25 b(=)g(\()p Fl(\000)p Fm(x)p Fo(\))1291 3407 y Fk(i)1345 3393 y Fo(=)g Fl(\000)p Fm(\037)1569 3412 y Fj(f)p Fk(i)p Fj(g)1667 3393 y Fo(\()p Fm(x)p Fo(\).)42 b Fb(2)0 3562 y Fo(This)30 b(suggests)j(the)g(follo)m(wing)d (mec)m(hanism)i(to)h(force)g(the)f(tables)g Fm(A)g Fo(to)h(ha)m(v)m(e) 2787 3539 y(^)2763 3562 y Fm(A)2831 3577 y Fk(\036)2905 3562 y Fo(=)28 b(0:)45 b(In)31 b(the)i(pro)s(of)0 3675 y(w)m(e)d(don't)g(ask)g(for)g(the)g(whole)f(table,)h(but)g(for)f(eac)m (h)i Fm(x)p Fo(,)f(w)m(e)h(ask)f(for)g(exactly)g(one)g(of)g(the)g(en)m (tries)g Fm(A)p Fo(\()p Fm(x)p Fo(\))0 3788 y(or)d Fm(A)p Fo(\()p Fl(\000)p Fm(x)p Fo(\).)40 b(Supp)s(ose)25 b(w.l.o.g.)40 b(that)28 b(w)m(e)f(ask)m(ed)h(for)f(the)g(bit)f Fm(A)p Fo(\()p Fm(x)p Fo(\).)40 b(When)27 b(w)m(e)h(need)f(the)g(bit)f Fm(A)p Fo(\()p Fm(x)p Fo(\),)i(w)m(e)0 3900 y(read)j(it)g(from)g(the)h (pro)s(of)e(and)h(when)f(w)m(e)i(need)f(the)h(bit)e Fm(A)p Fo(\()p Fl(\000)p 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(capture)h(NP-hard)e(problems.)38 b(This)25 b(is)h(b)s(ecause)h(in)f (this)g(case)i(one)0 4956 y(can)j(alw)m(a)m(ys)g(\014nd)d(a)j(correct)h (pro)s(of)d(for)h(a)h(YES)f(instance)g(through)g(Gaussian)f (elimination.)0 5181 y Fc(Implications:)50 b Fo(The)35 b(3)h(bit)f(PCP)f(has)i(the)f(follo)m(wing)f(corollaries.)56 b(The)35 b(\014rst)f(is)h(immediate)f(from)0 5294 y(the)27 b(PCP)-8 b(,)27 b(w)m(e)h(sho)m(w)f(a)h(gadget-based)g(reduction)e(for) h(the)h(second,)g(the)f(rest)g(are)h(obtained)f(b)m(y)g(similar)0 5407 y(reductions)i(\(see)i(H)-11 b(\027)-57 b(astad's)32 b(pap)s(er\).)p eop %%Page: 6 6 6 5 bop 3555 100 a Fo(6)0 399 y Fg(Theorem)33 b(6)0 511 y Ff(MAX-3-LIN)41 b(is)e Fo(\()669 476 y Fh(1)p 669 491 36 4 v 669 543 a(2)741 511 y Fo(+)26 b Fm(\017)p Fo(\))p Ff(-hard)40 b(to)g(appro)m(ximate,)j(i.e.)69 b(giv)m(en)40 b(an)f(instance)h(of)g(MAX-3-LIN,)i(it)d(is)0 624 y Fm(N)10 b(P)j Ff(-hard)30 b(to)h(tell)e(if)h Fm(O)s(P)13 b(T)37 b Fl(\025)25 b Fo(1)c Fl(\000)f Fm(\017)30 b Ff(or)h Fm(O)s(P)13 b(T)37 b Fl(\024)1752 589 y Fh(1)p 1752 604 V 1752 656 a(2)1818 624 y Fo(+)20 b Fm(\017)p Ff(.)0 812 y Fg(Theorem)33 b(7)0 925 y Ff(MAX-3-SA)-8 b(T)32 b(is)d Fo(\()669 889 y Fh(7)p 669 904 V 669 956 a(8)735 925 y Fo(+)20 b Fm(\017)p Fo(\))p Ff(-hard)30 b(to)h(appro)m(ximate.)0 1137 y Fg(Pr)n(oof:)70 b Fo(Consider)43 b(the)j(follo)m(wing)e (reduction)g(from)h(MAX-3-LIN)i(to)f(MAX-3-SA)-8 b(T.)47 b(Giv)m(en)e(an)0 1250 y(instance)f(of)h(MAX-3-LIN,)h(obtain)f(an)f (instance)g(of)h(MAX-3-SA)-8 b(T)46 b(b)m(y)f(ha)m(ving)f(the)h(same)g (set)g(of)0 1363 y(v)-5 b(ariables,)28 b(and)g(ha)m(ving)g(4)h(clauses) g(for)f(eac)m(h)i(equation.)40 b(If)28 b(the)h(equation)f(is)g Fm(x)17 b Fl(\010)f Fm(y)k Fl(\010)c Fm(z)30 b Fo(=)25 b(0,)k(then)g(the)0 1476 y(corresp)s(onding)23 b(4)j(clauses)f(are:)45 b(\026)-51 b Fm(x)10 b Fl(_)17 b Fo(\026)-52 b Fm(y)14 b Fl(_)h Fo(\026)-50 b Fm(z)t Fo(,)58 b(\026)-51 b Fm(x)10 b Fl(_)g Fm(y)k Fl(_)c Fm(z)t Fo(,)52 b Fm(x)10 b Fl(_)16 b Fo(\026)-51 b Fm(y)13 b Fl(_)d Fm(z)t Fo(,)27 b(and)50 b Fm(x)10 b Fl(_)g Fm(y)k Fl(_)h Fo(\026)-50 b Fm(z)t Fo(.)39 b(Consider)24 b(an)m(y)i Fl(f)p Fo(0)p Fm(;)15 b Fo(1)p Fl(g)0 1589 y Fo(assignmen)m(t)30 b(to)i(the)f(v)-5 b(ariables)29 b(of)i(the)g(MAX-3-LIN)h(instance.)41 b(The)30 b(corresp)s(onding)f(assignmen)m(t)h(to)0 1702 y(the)37 b(v)-5 b(ariables)35 b(of)i(the)f(MAX-3-SA)-8 b(T)38 b(instance)e(is)g(from)g Fl(f)p Fo(F)-8 b(alse,)39 b(T)-8 b(rue)p Fl(g)p Fo(,)39 b(with)c(0)i(mapp)s(ed)e(to)i(F)-8 b(alse)0 1815 y(and)42 b(1)i(mapp)s(ed)e(to)h(T)-8 b(rue.)79 b(Then)42 b(one)h(can)g(c)m(hec)m(k)i(that)f(if)e(a)h(setting)g(of)h (the)f(v)-5 b(ariables)41 b(satis\014es)0 1928 y(an)35 b(equation,)i(then)e(the)h(corresp)s(onding)d(setting)j(satis\014es)e (all)h(4)h(clauses.)55 b(But)36 b(if)e(a)i(setting)f(of)h(the)0 2041 y(v)-5 b(ariables)37 b(do)s(es)h(not)h(satisfy)f(an)g(equation,)j (then)d(the)h(corresp)s(onding)e(setting)h(can)h(satisfy)f(only)g(3)0 2154 y(out)f(of)h(the)f(4)h(clauses.)61 b(This)35 b(implies)g(that)j (if)e(the)h(optim)m(um)f(of)i(the)f(MAX-3-LIN)i(instance)e(is)f(at)0 2267 y(least)31 b(1)21 b Fl(\000)f Fm(\017)p Fo(,)31 b(then)g(the)g(optim)m(um)f(of)h(the)g(MAX-3-SA)-8 b(T)32 b(instance)e(is)g(also)h(at)h(least)f(1)21 b Fl(\000)f Fm(\017)p Fo(,)31 b(and)f(if)g(the)0 2379 y(optim)m(um)23 b(of)h(the)h(MAX-3-LIN)g(instance)f(is)f(at)i(most)1900 2344 y Fh(1)p 1900 2359 V 1900 2411 a(2)1953 2379 y Fo(+)8 b Fm(\017)p Fo(,)25 b(then)e(the)i(optim)m(um)e(of)h(the)g(MAX-3-SA)-8 b(T)0 2492 y(instance)30 b(is)f(at)j(most)789 2457 y Fh(7)p 789 2472 V 789 2524 a(8)855 2492 y Fo(+)20 b Fm(\017)p Fo(.)40 b Fb(2)0 2730 y Fg(Theorem)33 b(8)0 2843 y Ff(MAX-CUT)e(is)e Fo(\()616 2807 y Fh(16)p 616 2822 71 4 v 616 2874 a(17)717 2843 y Fo(+)20 b Fm(\017)p Fo(\))p Ff(-hard)30 b(to)h(appro)m(ximate.)0 3030 y Fg(Theorem)i(9)0 3143 y Ff(MAX-2-SA)-8 b(T)32 b(is)d Fo(\()669 3107 y Fh(21)p 669 3122 V 669 3175 a(22)770 3143 y Fo(+)20 b Fm(\017)p Fo(\))p Ff(-hard)30 b(to)h(appro)m(ximate.)0 3356 y Fg(Theorem)i(10)0 3469 y Ff(V)-8 b(ertex)32 b(Co)m(v)m(er)f(is)e Fo(\()687 3433 y Fh(7)p 687 3448 36 4 v 687 3500 a(6)753 3469 y Fl(\000)20 b Fm(\017)p Fo(\))p Ff(-hard)30 b(to)h(appro)m (ximate.)0 3755 y Fa(References)0 3958 y Fo([1])46 b(Bellare,)28 b(Goldreic)m(h)e(and)g(Sudan.)g(F)-8 b(ree)28 b(bits,)e(PCPs)g(and)h (non-appro)m(ximabilit)m(y)d({)j(to)m(w)m(ards)h(tigh)m(t)141 4071 y(results.)i(SIAM)g(Journal)f(on)h(Computing,)f(27\(3\):804{915,) 36 b(1998.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF