Graph Expansion and the Unique Games Conjecture

Transcription

1 Graph xpasi ad the ique Games Cjecture rasad Raghavedra MSR New glad Cambridge, MA David Steurer ricet iversity ricet, NJ ABSTRACT The edge expasi f a subset f vertices S V i a graph G measures the fracti f edges that leave S. I a d-regular graph, the edge expasi/cductace Φ(S) f a subset (S,V \S) d S S V is defied as Φ(S) =. Apprximatig the cductace f small liear sized sets (size δ) is a atural ptimizati questi that is a variat f the well-studied Sparsest Cut prblem. Hwever, there are kw algrithms t eve distiguish betwee almst cmplete edge expasi (Φ(S) = ε), ad clse t 0 expasi. I this wrk, we ivestigate the cecti betwee Graph xpasi ad the ique Games Cjecture. Specifically, we shw the fllwig: We shw that a simple decisi versi f the prblem f apprximatig small set expasi reduces t ique Games. Thus if apprximatig edge expasi f small sets is hard, the ique Games is hard. Alteratively, a refutati f the GC will yield better algrithms t apprximate edge expasi i graphs. This is the first -trivial reverse reducti frm a atural ptimizati prblem t ique Games. der a slightly strger GC that assumes mild expasi f small sets, we shw that it is G-hard t apprximate small set expasi. O istaces with sufficietly gd expasi f small sets, we shw that ique Games is easy by extedig the techiques f [4]. Categries ad Subject Descriptrs F.. [Thery f Cmputati]: Aalysis f Algrithms ad rblem Cmplexity Numerical Algrithms ad rblems Supprted by NSF grats , , art f this wrk de while at Micrsft Research New glad. ermissi t make digital r hard cpies f all r part f this wrk fr persal r classrm use is grated withut fee prvided that cpies are t made r distributed fr prfit r cmmercial advatage ad that cpies bear this tice ad the full citati the first page. T cpy therwise, t republish, t pst servers r t redistribute t lists, requires prir specific permissi ad/r a fee. STOC 0, Jue 5 8, 00, Cambridge, Massachusetts, SA. Cpyright 00 ACM /0/06...$0.00. Geeral Terms Algrithms, Thery. Keywrds Graph xpasi, ique Games Cjecture, Hardess f Apprximati, Spectral prfile.. INTRODCTION The pheme f vertex ad edge expasi i graphs has bee a subject f itese study with applicatis pervadig almst all braches f theretical cmputer sciece. Frm a algrithmic stadpit, apprximatig expasi r lack theref (fidig gd cuts r separatrs) is a fudametal ptimizati prblem with umerus applicatis. Yet, the cmputatial cmplexity f detectig ad apprximatig expasi i graphs is t very well uderstd. Amg the tw tis f expasi, this wrk will ccer mstly with edge expasi. Fr simplicity, let us first csider the case f a d-regular graph G = (V, ). The edge expasi f a subset f vertices S V measures the fracti f edges that leave S. Frmally, the edge expasi Φ(S) f a subset S V ca be defied as, Φ G(S) = (S, V \ S) d S where (S, V \ S) detes the set f edges with e edpit i S ad the ther edpit i V \ S. The cductace r the Cheeger s cstat assciated with the graph G is the miimum f Φ(S) ver all sets S with at mst half the vertices, i.e., Φ G = mi Φ G(S). S / The defiitis f cductace f sets Φ G(S) ad the graph Φ G ca be exteded aturally t -regular graphs, ad fially t arbitrary weighted graphs. Fr the sake f simplicity, we restrict ur atteti t regular graphs here, ad defer the discussi f geeral weighted graphs t the full versi. Hecefrth, we will use the tati µ(s) t dete the rmalized set size µ(s) = S /. The prblem f apprximatig Φ G als referred t as the the uifrm Sparsest Cut (equivalet withi a factr f ), is amg the fudametal prblems i apprximati algrithms. ffrts twards apprximatig Φ G have led t a rich bdy f wrk with strg cectis t spectral techiques ad metric embeddigs. The first apprximati fr cductace was btaied by discrete aalgues f the Cheeger iequality [9] shw by,

2 Al-Milma [] ad Al []. Specifically, they shw the fllwig: Therem. (Cheeger s Iequality). If λ detes the secd largest eigevalue f the suitably rmalized adjacecy matrix f a graph G the, λ Φ G p ( λ ). Sice the secd eigevalue λ ca be efficietly cmputed, Cheeger s iequality yields a apprximati algrithm fr Φ G, ideed e that is used heavily i practice fr graph partitiig. Hwever, the apprximati fr Φ G btaied via Cheeger s iequality is pr i terms f a apprximati rati, especially whe the value f Φ G is small (λ is clse t ). A O(lg ) apprximati algrithm fr Φ G was btaied by Leight ad Ra [7]. Later wrk by Liial et al. [8] ad Auma ad Rabai [6] established a strg cecti betwee the Sparsest Cut prblem ad the thery f metric spaces, i tur spurrig a large ad rich bdy f literature. Mre recetly, i a breakthrugh result Arra et al. [5] btaied a O( lg ) apprximati fr the prblem usig semidefiite prgrammig techiques. Small Set xpasi. Nte that the Φ G is a fairly carse measure f edge expasi, i that it is the wrst case edge expasi ver sets S f all sizes. I a typical graph (say a radm d-regular graph), smaller sets f vertices expad t a larger extet tha sets with half the vertices. Fr istace, all sets S f δ vertices i a radm d-regular graph have Φ(S) d with very high prbability, while the cductace Φ G f the etire graph is rughly. Mrever, the strger expasi exhibited by small sets has umerus applicatis i graph thery. A atural fier measure f the edge expasi f a graph is its expasi prfile. Specifically, fr a regular graph G the expasi prfile is give by the curve Φ G(δ) = mi Φ(S) δ [0, /]. µ(s)=δ The prblem f apprximatig the expasi prfile is seemigly far-less tractable tha apprximatig Φ G itself. Fr istace, there is kw algrithm fr the fllwig easily stated decisi prblem ccerig the expasi prfile: rblem. Gap-Small-Set xpasi (η, δ) Give a graph G ad cstats η, δ > 0, distiguish whether Φ G(δ) η r Φ G(δ) η. Spectral techiques fail i apprximatig the expasi f small sets i graphs. O e had, eve with the largest pssible spectral gap, the Cheeger s iequality cat yield a lwer bud greater tha / fr the cductace Φ G(δ). Mre imprtatly, there exists graphs such as hypercube where there are sets S f half the vertices with small cductace (Φ(S) < η), yet every sufficietly small set S satisfies Φ(S) η. This implies that Φ G (ad the secd eige value λ ) d t yield ay ifrmati abut Φ G(δ) fr small δ. I irregular graphs, it is mre cveiet t permit sets S withi a rage f sizes say [δ, 0δ], sice i a arbitrary (regular) graph there culd be sets satisfyig µ(s) = δ. ique Games Cjecture. The ique Games Cjecture (GC) f Kht [3] is amg the cetral pe prblems i hardess f apprximati, ad has fueled may develpmets i the area i recet years. The GC is shw t imply ptimal iapprximability results fr classic prblems like Max Cut [4], Vertex Cver [5] ad Sparsest Cut [6] ad cstrait satisfacti prblems [9]. Fr the sake f ccreteess, we frmally state the uique games cjecture here. The ique Games prblem is defied as fllws: Defiiti. A istace f ique Games represeted as Υ = (V,, Π, [R]) csists f a graph ver vertex set V with the edges betwee them. Als part f the istace is a set f labels [R] = {,..., R}, ad a set f permutatis π v w : [R] [R] fr each edge e = (w, v). A assigmet A f labels t vertices is said t satisfy a edge e = (w, v), if π v w(a(w)) = A(v). The bjective is t fid a assigmet A f labels that satisfies the maximum umber f edges. As is custmary i hardess f apprximati, e defies a gap-versi f the ique Games prblem as fllws: rblem. (ique Games (R, ε, η)) Give a ique Games istace Υ = (V,, Π = {π v w : [R] [R] e = (w, v) }, [R]) with umber f labels R, distiguish betwee the fllwig tw cases: ( ε)- satisfiable istaces: There exists a assigmet A f labels that satisfies a ε fracti f edges. Istaces that are t η-satisfiable: N assigmet satisfies mre tha a η-fracti f the edges. The ique Games Cjecture asserts that the abve decisi prblem is N-hard whe the umber f labels is large eugh. Frmally, Cjecture. (ique Games Cjecture [3]). Fr all cstats ε, η > 0, there exists large eugh cstat R such that ique Games (R, ε, η) is N-hard. While the implicatis f the cjecture are well-uderstd, there has bee much slwer prgress twards its resluti. Results supprtig the truth f GC have bee especially difficult t shw. I particular, it is ukw whether may f the implicatis f GC are equivalet t the cjecture. I ther wrds, it is etirely csistet with existig literature that all the implicatis f GC prblems like Max Cut ad Vertex Cver hld, but the cjecture itself is false. Mre precisely, althugh the ique Games prblem is kw t efficietly reduce t classic prblems like Max Cut ad Vertex Cver, there are kw reverse reductis frm these prblems back t ique Games. The ly reverse reducti twards which there is ay literature is the reducti frm Max Cut t ique Games. Nte that the Max Cut prblem is a special case f ique Games ver the biary alphabet {0, }. Hece a Max Cut istace is readily reduced t a ique Games istace by usig parallel repetiti. With a sufficietly strg parallel repetiti therem (cjectured by Feige et al. []), this wuld yield a reverse reducti frm Max Cut t ique Games. frtuately, a strg parallel repetiti therem f this ature was shw t t hld by Raz [4]. Subsequet wrk by Barak et al. [7] almst etirely ruled ut this apprach t reduce Max Cut t ique Games.

3 xpasi ad ique Games. Vertex ad edge expasi i graphs appear t be clsely tied t the hard istaces fr liear ad semidefiite prgrammig relaxatis. May itegrality gap istaces have bee cstructed fr prblems like Vertex Cver r Max Cut agaist the liear prgrammig hierarchies such as Lvász Schriver ad Sherali Adams hierarchies (see [8, 5] ad the refereces therei). Nt ly d mst f these istaces csist f expadig graphs but the argumets rely crucially either vertex r edge expasi. The situati is little bit mre subtle i case f semidefiite prgrammig. Semidefiite prgrams ca apprximate Max Cut well istaces that have very gd cductace (spectral gap). Hece, the SD itegrality gap istaces kw are graphs where small sets expad well, while the larger sets d t. Ideed, SD itegrality gap cstructis fr Max Cut [0, 6, ], Vertex Cver [], ique Games [6, 0] ad Sparsest Cut [6] all have ear-perfect edge expasi fr small sets. I case f ique Games, t ly d all kw itegrality gap istaces have ear-perfect edge expasi f small sets, eve the aalysis relies directly this prperty. Furthermre, it is kw that the best pssible sudess fr ique Games with label size R ad cmpleteess ε is a cstat η(r, ε) which is rughly. The cstat R ε/ η arises directly frm the expasi f sets f size i a R certai graph defied ver the Gaussia space. While this suggests that ique Games is clsely tied t expasi f small sets i graphs, smewhat ctrastigly, Arra et al. [4] shw that ique Games is easy whe the cstrait graph ivlved is a gd spectral expader, i.e., has a -trivial spectral gap fr the Laplacia. Mtivated by the abve reass, we ivestigate the cecti betwee graph expasi ad ique Games i this wrk.. Results The mai result f this wrk is a reducti frm the prblem f apprximatig expasi f small sets t the ique Games prblem. The mai implicati f the result ca be succictly stated as fllws. Let us csider the fllwig hardess assumpti abut the cmplexity f the Gap-Small-Set xpasi prblem. Cjecture.3. (Gap-Small-Set xpasi Cjecture) Fr every η > 0, there exists δ such that the prblem Gap- Small-Set xpasi (η, δ) is N-hard. The, a immediate csequece f the reducti preseted i this wrk is, Therem.4. The Gap-Small-Set xpasi cjecture implies the ique Games Cjecture. T the best f ur kwledge, this is the first -trivial reverse reducti frm a atural cmbiatrial ptimizati prblem t ique Games. O e had, it cects the smewhat -stadard prblem f ique Games t the much well-studied prblem f apprximatig graph expasi. Furthermre, the result makes ccrete the cspicius presece f small set expasi i SD itegrality gaps fr ique Games ad related prblems. While a cfirmati f GC was kw t imply ptimal iapprximability results fr fudametal prblems, a refutati f GC was t kw t frmally imply ay ew algrithmic techique. A imprtat implicati f the abve result is that a refutati f the GC wuld yield a algrithm fr apprximatig edge expasi (ly i a certai regime) i graphs a basic ptimizati prblem. Nw we state the mai result f the paper frm this algrithmic stadpit. T this ed, we frmally state a hypthesis that wuld be emerge frm a refutati f GC. Hypthesis.5 (ique Games is easy). There exists a cstat ε > 0 ad a fucti f : N N such that give a ique Games istace Υ with vertices ad k labels, it is pssible distiguish betwee the cases pt(υ) ε ad pt(υ) ε i time f(k). Therem.6. Suppse the abve hypthesis ( ique Games is easy) hlds fr a cstat ε 0 ad a fucti f, the there exists a fucti g : [0, ] N such that give ay graph G with vertices, ad it is pssible t distiguish whether Φ G(δ) ε r Φ G(δ) > ε fr sme abslute cstat ε i time g(δ). A smewhat strger csequece f the ique Games is easy hypthesis fllws usig the parallel repetiti therem. The fllwig crllary is a immediate csequece f the parallel repetiti therem f Ra [3] ad ur reducti frm Gap-Small-Set xpasi t ique Games. Crllary.7. Suppse the hypthesis ique Games istace is easy) hlds fr a cstat ε 0 ad a fucti f. The, give a graph G with vertices ad parameters ε, δ such that ε < ε fr sme abslute cstat ε, we ca distiguish the fllwig cases i time g(ε,δ) fr sme g : [0, ] N.. There exists S V with µ(s) = δ ad Φ(S) ε. very set S V with µ(s) 500δ/ε satisfies Φ(S) 500 ε... Twards a quivalece A atural questi that arises frm Therem.4 is whether the ique Games Cjecture is equivalet t the Gap- Small-Set xpasi cjecture. Mre specifically, culd it be true that ique Games Cjecture implies the Gap- Small-Set xpasi cjecture. Shwig a result f this ature amuts t btaiig a reducti frm ique Games t Gap-Small-Set xpasi. Despite the large umber f G reductis ad a fairly thrugh uderstadig f hw t cstruct reductis frm ique Games, btaiig reductis t graph expasi prblems is fte prblematic. The mai issue is that hardess reductis via lcal gadgets d t alter the glbal structure f the graph. Fr example, if the ique Games istace is discected, the resultig graph prduced by a gadget reducti is als discected. Hece, t shw a G-hardess result fr a glbal prperty such as expasi, it fte seems ecessary t assume the crrespdig glbal prperty the cstrait graph f the ique Games. Mre specifically, i ur case we require that the ique Games istace has gd lcal expasi, i that sufficietly small sets have cductace clse t. The frmal statemet f the mdified ique Games Cjecture with mild expasi small sets is give belw:

4 Cjecture.8. (ique Games with Small Set xpasi Cjecture) Fr every ε > 0, there exists δ such that fr all η > 0 the fllwig prblem is N-hard with a sufficietly large R = R(ε, η): Give a istace Υ = (V,, Π = {π v w : [R] [R]}, [R]) f ique Games with alphabet size R R 0 distiguish betwee the fllwig cases: Cmpleteess There exists a assigmet A : V [R] that satisfies at least ε fracti f edges. Sudess very assigmet A : V [R] satisfies at mst η fracti f edges ad the cstrait graph (V, ) satisfies the fllwig expasi prperty: (Lcal xpasi rperty) Fr every set S with µ(s) δ, Φ(S) ε. The abve cjecture assumes fairly mild expasi i that sufficietly small sets have cductace clse t. I fact, existig SD itegrality gap istaces fr ique Games (see [6, 0]) satisfy the abve lcal expasi prperty. der this strger ique Games Cjecture stated abve, we shw the fllwig hardess result fr Gap-Small-Set xpasi. Therem.9. The ique Games with Small Set xpasi cjecture implies the Gap-Small-Set xpasi cjecture. As a immediate csequece f the G hardess reducti we als btai super-cstat hardess f apprximati fr the k-desest Subgraph rblem. Therem.0. Assumig the ique Games with Small Set xpasi cjecture, it is N-hard t apprximate the k-desest Subgraph rblem t ay cstat factr. sig stadard techiques, the G-hardess reductis ca be used t cstruct SD itegrality gaps fr the Gap- Small-Set xpasi ad the k-desest Subgraph rblem. Due t space cstraits, we defer the prfs f Therem.9 ad Therem.0 t the full versi.. ique Games with Small Set xpasi are asy Further explrig the cecti betwee ique Games ad small set expasi, we shw that the ique Games prblem is easy whe there is sufficietly high expasi f small sets. This result geeralizes the wrk f [4] which shwed that ique Games is easy whe the assciated cstrait graph is a gd spectral expader. We will use a aalytic ti f a graph beig a small set expader fr this purpse. This is aki t spectral expasi f a graph istead f cmbiatrial expasi. Defiiti. Fr a graph G ad δ > 0, defie Λ G(δ) as the miimum f X u X v (u,v) G ver all cllecti f vectrs {X v} v V that satisfy the cstraits Xu, Xv δ, (.) u,v V Xu =. (.) like the cmbiatrial small set expasi Φ G(δ), the quatity Λ G(δ) is efficietly cmputable, thus makig it easy t recgize graphs that satisfy the required expasi prperty. Raghavedra et al. [] shw that this parameter apprximates the cmbiatrial expasi prfile Φ G(δ) as fllws: Λ G(δ) Φ G(δ) O ΛG(δ/) lg(/δ) /. I this wrk, we shw that ique Games is easy if the cstrait graph G has sufficietly large Λ G(δ). The frmal statemet f the result is as fllws: Therem.. Let Υ be uique game with SD value ε ad cstrait graph G. Suppse there exists δ > 0 such that Λ G(δ) > 0 6 ε lg( /εδ), the the itegral value f Υ is at least δ/0. Crllary.. If the cstrait graph G assciated with a uique games istace Υ satisfies Λ G(δ) > 0 6 ε lg( /εδ), the it is easy t distiguish whether istace has value ε r less tha δ. Idepedet f this wrk, Arra et al. [3] als shw that ique Games is easy if the uderlyig cstrait graph has sufficiet lcal expasi usig differet techiques. Due t space cstraits, we defer the prf f Therem. t the full versi. Relati with Cjecture.8: Fr δ < /ε the quatity 0 6 ε lg( /εδ) >, while Λ G(δ) fr all graphs G ad all δ > 0. Hece, Therem. is t applicable graphs with expasi f sets f size δ < /ε. I Cjecture.8, there is lwer bud the size δ f sets with lcal expasi. Csequetly, Therem. ly yields a upper bud the chice f the set size δ i Cjecture.8.. RLIMINARIS Ntati: less therwise stated, all graphs csidered hecefrth ca be assumed t be regular uweighted graphs. The prfs ad results geeralize t arbitrary weighted graphs, but we defer its discussi t the full versi. Give a graph G = (V, ), we will use v V t dete a vertex sampled frm the prbability distributi give by degrees deg(v) (i this case, the uifrm distributi). Furthermre, fr a set S V, µ(s) = v V {v S}. Similarly, the tati e will dete a edge sampled frm a distributi prprtial t the weights (the uifrm distributi i uweighted graphs). We dete by (A, B) the set f edges with e edpit i A ad the ther edpit i B. Fr a subset f edges, will dete the sum f the edge weights withi. Therefre fr a subset f edges, e [e ] = / =. Defiiti 3. Fr a vertex subset S V, we defie (S) def = (S, V \ S) ad µ(s) def = X s S deg(s). Defiiti 4. (Cductace) The cductace/cheeger s cstat assciated with a subset S V is give by Φ(S) = (S). The cductace/cheeger s cstat fr the graph G µ(s) is Φ G = mi µ(s) Φ(S).

5 Due t space cstraits we mit the prfs f the fllwig simple facts. Fact.. Fr all a, b [0, ], a b a + b 4ab. Lemma. (Glrified Markv Iequality). Let Ω be a prbability space ad let X, Y : Ω R + be tw jitly distributed -egative radm variables ver Ω. Suppse X γ Y. The, there exists ω Ω such that X(ω) γy (ω) ad Y (ω) Y/. artial -prver games: A geeral -prver game Γ is specified by a vertex set V, a edge set, a alphabet Σ, ad a cllecti f predicates {Π u,v} idexed by vertex pairs u, v V. The value f the game Γ is defied as the maximum success prbability uv (F (u), F (v)) Π u,v, ver all strategies F : V Σ. Give a -prver game Γ, a partial game is e i which the prvers are permitted t refuse t aswer the questi. The prvers wi ly if bth f them refuse t aswer the questi r else they bth aswer crrectly as per the game Γ. T preclude the trivial strategy that always refuses t aswer questis, we require that the prvers aswer at least a α- fracti f the questis fr sme cstat α > 0. Frmally, we defie the α-partial value f a -prver game Γ as the maximum success prbability uv (F (u), F (v)) Π u,v / F (u), ver all α-partial strategies F : V Σ { }. Here, is a desigated symbl (t a member f Σ) ad a α- partial strategy is a assigmet F : V Σ { } such that {F (u) } α. As usual, the tati v V meas that v is sampled with prbability prprtial t its degree. Orgaizati: I the remaider f the paper, we preset the reducti frm Gap-Small-Set xpasi t ique Games ad its aalysis. Fr cceptual clarity, we subdivide the reducti i t tw parts: I the first part (Secti 3), we reduce Gap-Small-Set xpasi t a partial ique games, ad the i Secti 4 exhibit a geeric reducti frm partial -prver games t a crrespdig traditial -prver game. Fially, i Secti 5 we wrap up the prf f the mai result f this wrk - Therem FROM GA-SMALL-ST-XANSION TO ARTIAL NIQ GAMS I this secti, we utlie the key ideas f the reducti frm Gap-Small-Set xpasi t artial ique Games. First Attempt. Let G = (V, ) be a istace f the Gap-Small-Set xpasi (η, δ) prblem. Fr the sake f simplicity, let G be a d-regular uweighted graph. We defie the fllwig uique game: Fix R =. The referee/verifier picks R edges M = δ {(u, v ),..., (u R, v R)} uifrmly at radm frm G. The referee seds a radm permutati f the tuple = (u,..., u R) t the first prver, ad a radm permutati f the tuple V = (v,..., v R) t secd prver. The prvers are expected t pick e f the vertices f the tuple they receive. Specifically, the aswer set f the first prver is {u,..., u R}, while the secd prver s aswer set is {v,..., v R}. The prvers wi if they pick tw vertices (u i, v i) crrespdig t sme edge i the set M. Cmpleteess: Let us suppse there exists a set S f δ vertices such that Φ(S) ε. I this case, the prvers ca use the fllwig simple strategy: If exactly e vertex frm S appears i the tuple, retur that vertex, else refuse t aswer the questi. The set S is f size δ ad a questi V R has δ vertices. Therefre, with cstat prbability (at least 0.0) exactly e f the vertices frm S appears i the questi, ad the prvers aswer the questi. Csequetly, the abve strategy is a valid partial strategy. Observe that if the set S has small cductace Φ(S) ε, a radm edge icidet S is with high prbability cmpletely ctaied withi S. I ther wrds, if u i S the with very high prbability its eighbur v i als belgs t S. This implies that wheever the first prver decides t aswer the vertex u i, the secd prver als aswers v i with very high prbability. Hece a small -expadig set S traslates directly i t a gd partial strategy fr the uique game. Sudess: Let F : V R [R] be a strategy fr the tw prvers that succeeds with prbability at least. Fr a vertex sequece V R, let + i x V R dete the vertex sequece btaied by isertig x at idex i. Fr every vertex sequece V R ad a idex i [R], let us defie a {0, }-valued fucti F (i) as fllws: Specifically, F (i) F (i) (x) = if F ( +i x) = i else it is 0 is the idicatr fucti f the set f vertices x such that the strategy decides t pick up the vertex x, whe it is iserted at the i th lcati i. Ntice that, fr the strategy prpsed i the cmpleteess case, fr every settig f, the set F (i) is either the expadig set S r the empty set. xtraplatig frm here, it is atural t lk fr -expadig sets by csiderig the set F (i) fr a a radm chice f V R ad i. This is the basic ituiti behid the sudess aalysis preseted i Secti 3.. Over a radm chice f V R ad i, the expected size f F (i) is ideed rughly Θ( ) = Θ(δ). Hwever, it R culd be pssible that with very high prbability ver the chice f ad i, F (i) is either t large r t small a set. T rule ut this pssibility, we mdify the abve uique game by icludig radm ise i t the questis. Mre specifically, the referee chages each vertex f the questis, V idepedetly with prbability ε, befre sedig it t the prvers. I the mdified game, we bud the secd mmet f the radm variable the size f F (i) radm chice f ad i. ver a Frmal Reducti. Here we preset the details f the reducti frm Gap- Small-Set xpasi prblem t artial ique Games. Fr a vertex v V ad ε > 0, we defie a distributi N ε(v) V as fllws: with prbability ε, we utput u := v ad with prbability ε, we utput u V. Fr a vertex

6 sequece v,..., v R, the tati N ε(v,..., v R) refers t the prduct distributi N ε(v ) N ε(v R) V R. Fr a edge e, we defie D(e) t be the distributi that picks a radm edpit f e (uifrmly). Fr a edge sequece e,..., e R, the distributi D(e,..., e R) V R is the prduct f the distributis D(e ),..., D(e R). Fr a permutati π : [R] [R] ad a sequece A V R, we write A = π.a t dete the permutati f A accrdig t π, i.e., A π(i) = A i fr all i [R]. Let R N. We defie a ique Games istace Υ = Υ R,ε(G) with vertex set V R ad alphabet Σ = [R]. The edge cstraits i Υ are give by the tests perfrmed by the fllwig verifier. SS-t-G Reducti Let F : V R Σ be a assigmet t the istace Υ. The value f the assigmet is the success prbability f the fllwig test:. Sample R radm edges e,..., e R frm ad let M := (e,..., e R).. Sample A D(M) ad B D(M). 3. Sample à Nε(A) ad B N ε(b). 4. Sample tw permutatis π, π : [R] [R]. 5. Output Success if F (π.ã) π = π F (π. B). (3.) I the remaider f the secti, we will shw the fllwig therem. Therem 3.. Fr η <, give a graph G = (V, ), the 4 3 reducti prduces a istace Υ f ique Games such that: Cmpleteess: If Φ(S) η fr sme S with µ(s) ˆ, 0R R the there exists a /0e-partial strategy f value at least 40eη 4ε. Sudess: If there exists a α-partial strategy with value η fr the ique Games istace Υ, thehthere exists i a set S V with Φ(S) 96η ad µ(s). α, 6 4R εηr Due t space limitatis, we defer the smewhat straightfrward prf f the cmpleteess case t the full versi. The sudess claim f the abve therem is prve i the ext secti (Lemma 3.3). 3. Sudess Let F : V R [R] { } be a partial assigmet fr the uique game Υ. Fr V R ad x V, we let f(, x) [0, ] dete the prbability that F selects the crdiate f x after we place it at a radm psiti f ad permute the sequece radmly, i.e., f(, x) def = i [R] π {F (π.( + i x)) = π(i)}. (3.) Here, + i x detes the vertex sequece i V R btaied frm by isertig x as the i-th crdiate (the rigial crdiates i,..., R f are mved by e t the right). Fr M R, we defie a fucti f M : V [0, ] as fllws f M (x) def = D(M) f(ũ, x). (3.3) (Ũ, x) Nε(,x) We establish the fllwig three relatively straight-frward prperties f the fuctis f M. rpsiti 3.. Suppse that F has value η fr the game Υ ad that F differs frm exactly a α fracti f the iputs, i.e., α := X V R {F (X) }. The the fllwig hld, M R M R f M (x) = α R, (3.4) fm (x) M εr R (3.5) e x D(e) fm (x) = ( η) α. (3.6) R Befre prvig the prpsiti, we will first shw that the existece f such fuctis {f M } as utlied i the prpsiti, suffices t fiish the sudess aalysis. Lemma 3.3. Suppse there exists a cllecti f fuctis {f M } satisfyig the three cditis f 3. fr sme η. 4 The there exists a set S V with µ(s) ˆ α, 6 4R εr ad Φ(S) 96η rf. Let X ad Y be the fllwig -egative radm variables ver the prbability space R, X(M) := f M (x)f M (y) Y (M) := f M (x). xy Sice X Y (pitwise), X = Y X is als a egative radm variable ver R. The cditis (3.6) ad (3.4) imply that X η Y ad Y = α /R. Hece, Lemma. (Glrified Markv Iequality) asserts that there exists a edge sequece M R such that X (M ) 4ηY (M ) ad Y (M ) Y/. I ther wrds, fm (x)fm (y) ( 4η) fm (x), (3.7) xy f M (x) α. (3.8) R By cditi (3.5), we ca als upper bud the expectati f f M, fm (x). (3.9) εr Lemma 3.4 shws that we ca rud the fucti f M t a cut S that satisfies Φ(S ) 96η, ad µ(s ) ˆ α 4R, 6 εr. Lemma 3.4. Let G be a graph with vertex set V ad edge set. Suppse f : V [0, ] is a buded fucti V such that f(x)f(y) ( η) f(x), (3.0) xy fr η <. Let β := f(x). The, there exists a vertex 6 set S V such that µ(s ) ˆ β, 3β ad Φ(S ) 4η. rf. Defie a distributi ver sets S give by the fllwig samplig prcedure:

7 . sample a threshld t [ /9, 4 /9] uifrmly at radm,. utput the set S := x f (x) > t. Sice f = β the set S always satisfies µ(s) [f] /3 Nte that cditi (3.0) implies ( η)β = 3β. (3.) f(x)f(y) `f (x)+f (y) = f. xy xy Therefre we have f( f) ηβ. previus iequality, ηβ [f < /3] [f( f) f < /3] Rewritig the 3 [f < /3] [f f < /3]. This implies a lwer bud the size f the set S as fllws, µ(s) [f /3] [f /3] [f f /3] = [f] [f < /3] [f f < /3] β( 3η). (3.) Fially, we estimate the budary f the set S. (S) = [ S(x) S(y) ] ˆ3 f (x) f (y) xy xy x f xy f(x)f(y) ( 0 f ad Fact.) ηβ (3.3) The claim fllws frm (3.),(3.) ad (3.3). 3. rf f rpsiti 3. Let F : V R [R] { } be a partial strategy f value η fr the game Υ. Suppse F differs frm exactly a α fracti f iputs, i.e., X V {F (X) [R]} = α. We prve the three cditis f 3. i the fllwig lemmas. Lemma 3.5. M R f M (x) = α R. rf. We urll the defiiti f f M (x), M R = f M (x) M R, D(M) i [R],(Ũ, x) Nε(,x) π = {F (X) = r} = α. X V R R, r [R] F (π.(ũ +i x))) = π(i) I the last equality, we use that the jit distributi f π.(ũ +i x) ad π(i) is the same as the jit distributi f X ad r. Lemma 3.6. M R e x D(e) f M(x) = ( η) α R. rf. Fr a edge sequece M R ad r [R], the tati M r refers t the edge sequece i R btaied frm M by remvig the r-th crdiate. Similarly, M r detes the r-th crdiate f M. Recall that F is a partial strategy with value ( η) α the game Υ. xpressig this fact we get, ( η) α R = M R,A,B D(M) Ã N ε(a), B Nε(B) r [R] = M R, r [R] = M R, r [R] = M R, r [R] = M R, r [R] = M R A D(M) F (π.ã) = π(r) F (π. B) = π π,π (r) x D(M r) D(M r ) (Ũ, x) Nε(,x) F (π.ã) = π(r) Ã N ε(a) π x D(M r) D(M r ) x D(M r) π F (π.(ũ +r x)) = π(r) f M r (x) f M (x). e x D(e) f(ũ, x) (Ũ, x) Nε(,x) I the secd equality, we use the fact that the evets F (π.ã) = π(r) ad F (π. B) = π (r) are idepedet give M R ad r [R]. I the third equality, e r detes the edge i the r-th crdiate f M. Lemma 3.7. Suppse εr is sufficietly large. The fr every M R, f M (x). εr rf. As befre, we first urll the defiiti f f M (x) f M (x) = = D(M) D(M) (Ũ, x) Nε(,x) f(ũ + x, x) (Ũ, x) Nε(,x) F (π.(ũ +i x))) = π(i) i [R] π (3.4) T estimate this expectati, we geerate radm variables π.(ũ +i x) ad π(i) i a differet fashi. Let dete a placehlder symbl (t a member f V ). Csider the fllwig radmized prcedure geeratig A V R, r [R]:. Sample D(M), a permutati π : [R] [R] ad a idex i [R]. Set B := π.( + i ).. Geerate B (V { }) R by chagig each crdiate f B radmly t with prbability ε. Let I [R] dete the set f crdiates f B that ctai. 3. Geerate A V R by replacig all the placehlders i B by vertices sampled frm V. 4. Sample r I uifrmly at radm. We claim that the jit distributi f A ad r is the same as the jit distributi f π.(ũ +i x) ad π(i) i (3.4). By cstructi, the jit distributi f A ad π(i) is the same as the jit distributi f π.(ũ +i x) ad π(i) i (3.4) (with a bvius cuplig). Frm the descripti f the cstructi, it is als clear that the r-th crdiate f A has

8 the same distributi as the π(i)-th crdiate f A (bth are sampled frm V ). Sice the pair (A, r) is idetical i distributi t the pair (π.(ũ +i x), π(i)), we ca fiish the prf f the lemma as fllws fm (x) = = A,r = A, I = A, I D(M), (Ũ, x) Nε(,x) F (A) = r r I F (A) = r I {F (A) I} t Bim(R,ε).5 + εr t Bim(R,ε).5 εr + Ω(εR) εr i [R] π F (π.(ũ +i x))) = π(i) (by (3.4)) (A ad r geerated as abve) (A ad I geerated as abve) t+ ( I + Bim(R, ε)) {t + εr/.5} (usig Cherff bud) (fr sufficietly large εr) 4. FROM ARTIAL -ROVR GAMS TO TOTAL -ROVR GAMS We shw a geeral reducti frm a partial -prver game Γ t a crrespdig -prver game Γ. Recall that i the partial game Γ, the prvers are allwed t refuse t aswer questis. Hwever, i the game Γ such chice shuld be available. T achieve this, the referee gives multiple questis frm Γ simultaeusly ad the prvers have t aswer ay questi f their chice. Specifically, the questis i Γ csists f a sequece f c questis frm the game Γ. The prvers are required t chse e f the c questis t aswer, ad retur the aswer. The prvers wi ly if they pick the crrespdig pair f questis (say the i th questi), ad als aswer the questi crrectly as per the game Γ. Frmally, we will shw the fllwig therem i the remaider f the secti. Therem 4.. Fr all psitive itegers c, give a -prver game Γ with vertices, there is a reducti t ather - prver game Γ ruig i time O(c) such that: Cmpleteess If the α-partial value f Γ is at least ε, the the value f Γ is at least ε e α c. Sudess If the value f Γ is at least η, the the /3c-partial value f Γ is at least η. Furthermre, the reducti preserves the uiqueess prperty f the games. We will assume that the edge set f the game Γ des t ctai ay self-lps. We dete by the fllwig distributi ver vertex pairs (u, v) V : sample a radm edge e ad let u ad v be (idepedet) radm edpits f e (i particular, u = v with prbability /). We add predicates Π u,u = {(a, a) a Σ} fr all self-lps (u, u) V t the cllecti {Π u,v}. Fr a parameter c N, let Γ be the fllwig -prver game: Reducti frm a partial -prver game Γ t a ttal -prver game Γ.Sample c vertex pairs (u, v ),..., (u c, v c)..sed the vertex sequece u,..., u c t the first prver ad the sequece v,..., v c t the secd prver. 3.Let (i, a) ad (j, b) be their aswers. 4.The prvers wi if i = j ad (a, b) Π ui,v j. We bserve that the reducti preserves the uiqueess prperty. Observati 4.. If Γ is a uique game, the Γ is a uique game as well. We will shw Therem 4. i tw parts: Lemma 4.3 ad Lemma 4.4. Lemma 4.3 (Cmpleteess). If the α-partial value f Γ is at least ε, the the value f Γ is at least ε e α c. rf. Let F : V Σ { } be a α-partial strategy f value ε. We csider the fllwig strategy F fr Γ. ( F (i, a) if (F (u i) Σ) ( j < i, F (u j) = ), (u,..., u c) := (, ) if F (u ) =... = F (u c) =. (I wrds, the prver aswers with the first aswer i the list F (u ),..., F (u c) (igrig ). If the partial strategy refuses t aswer all iputs u,..., u c, the the prver returs a arbitrary aswer.) Let u v,..., u cv c be a sequece f c vertex pairs, idepedetly draw frm. The prbability f the evet F (u ) =... = F (u c) is at mst ( α) c e αc. Let us cditi the cmplemetary evet, i.e., the evet that F (u i) fr at least e crdiate i [c]. Let i 0 be the first crdiate such that F (u i0 ) r F (u i0 ). The wiig prbability f the prvers (cditied the evet that F (u i) fr at least e crdiate i [c]) is equal t (F (u i0 ), F (v i0 )) Π ui0,vi0 = (F (u), F (v)) Π u,v F (u) Σ F (v) Σ uv {(F (u), F (v)) Πu,v} uv {F (u) Σ F (v) Σ} = uv = uv {(F (u), F (v)) Π u,v} {F (u) Σ} uv {F (u), F (v) Σ}. Withut lss f geerality, we may assume {F (u) Σ} = α. Sice the partial strategy F has value ε, we have uv {(F (u), F (v)) Π u,v} / ( ε)α+ / α. Therefre, als uv {F (u), F (v) Σ} ( ε /)α. We ca cclude (F (u), F (v)) Π u,v F (u) Σ F (v) Σ uv ( ε/)α (+ε/)α ε. It fllws that the value f the strategy F fr the game Γ is at least ε e αc. Lemma 4.4 (Sudess). If the value f Γ is at least η, the the /3c-partial value f Γ is at least η.

9 rf. Let F : V c [c] Σ be a strategy fr Γ f value η. We first cstruct a partial strategy fr Γ that uses shared radmess. Fr i [c] ad u V c, we defie a partial strategy F i,u : V Σ { } as ( a if F (u + i u) = (i, a), F i,u(u) := therwise. Here, u + i u detes the vertex sequece u V c btaied frm u by isertig u as the i-th crdiate (the rigial crdiates i,..., c f u are mved by e t the right). It is clear that i [c], u V c, F i,u(u) Σ = /c. (4.) Sice the value f F is η, we have ( η)/c = (F i,u(u), F i,v(v)) Π u,v = i [c], uv c + i [c], uv c i [c], uv c, uv,, uv (F i,u(u), F i,v(v)) Π u,v F i,u(u) = F i,v(u) Σ Frm (4.), it fllws that bth prbabilities i the previus equati are at mst /c. Hece, fr their average t be ( η)/c, bth f them have t be at least ( η)/c, (F i,u(u), F i,v(v)) Π u,v i [c], uv c i [c], uv c, uv, ( η)/c, (4.) F i,u(u) = F i,v(u) Σ ( η)/c. (4.3) T further aalyze the partial strategies, we defie tw radm variables Vl(i, u, v) := F i,u(u) Σ F i,v(u) Σ, Val(i, u, v) := (F i,u(u), F i,v(v)) Π u,v. uv The measure (i, u, v) is as fllws: We chse i [c] uifrmly at radm, ad sample uv frm c. It is clear that Val Vl (pitwise) ad h Vl = {Fi,u(u) Σ} (4.4) i [c], uv c + {F i,v(u) Σ} = /c i [c], uv c, i F i,u(u), F i,v(u) Σ F i,u(u), F i,v(u) Σ ( + η)/c (usig (4.3)). (4.5) O the ther had, sice the value f F is η, Val is give by, (F i,u(u), F i,v(v)) Π u,v = ( η)/c. i [c], uv c, uv (4.6) At this pit, we culd use Lemma. (Glrified Markv Iequality) t argue that there exists a triple (i, u, v) such that Val(i, u, v) ( O(η))Vl(i, u, v) ad Vl(i, u, v) Vl/ /c. Fr the reader s cveiece, we repeat the (shrt) argumet: Let η 3η be such that η = ( η)/( + η). Let Gd be the evet Val ( η )Vl ad let Bad be the cmplemetary evet. We dete by Gd ad Bad the idicatr variables f these evets. We ca relate Gd Vl ad Vl as fllws 0 Val ( η ) Vl (usig (4.5) ad (4.6)) Gd Vl + ( η ) Bad Vl ( η ) Vl = η Gd Vl η Vl. Hece, we ca fid i, u, ad v such that Gd (i, u, v) = ad Vl(i, u, v) Vl/, i.e., (F i,u(u), F i,v(v)) Π u,v uv ( η ) F i,u(u) Σ F i,v(u) Σ F i,u(u) Σ F i,v(u) Σ, (4.7) ( η)/c. (4.8) We claim that the tw cditis (4.7) ad (4.8) allws us t cstruct a O(/c)-partial strategy f value O(η). Mre ccretely, we ca mdify F i,u ad F i,v i such a way that F i,u(u) = F i,v(u) fr all u V while maitaiig the cditis (4.7) ad (4.8). (Here, we use the fact that with prbability /, we test F i,u(u) = F i,v(u) Σ.) I this way, we get a assigmet F : V Σ { } such that F (u) = if F i,u(u) = F i,v = ad F (u) {F i,u(u), F i,v(u)} Σ therwise. Furthermre, the assigmet F satisfies F (u) Σ (F (u), F (v)) Π u,v uv = F i,u(u) Σ F i,v(u) Σ Let α = {F (u) Σ}. We have ( η ) α, (F i,u(u), F i,v(v)) Π u,v. uv uv (F (u), F (v)) Π u,v = (F (u), F (v)) Π u,v + α uv It fllws that uv { (F (u), F (v)) Π u,v} ( 4η ) α. We ca cclude that F is a ( η) /c-partial strategy f value at least 4η η fr Γ. 5. WRAING The mai result f this wrk amely Therem.4 is a immediate csequece f the fllwig therem. Therem 5.. Give a regular graph G f size ad parameters ε, δ > 0 such that δ = m/ fr sme iteger m, we ca cmpute i plymial time a ique Games istace Υ with (/ε) O(/δ) variables ad with ply(/εδ) labels such that the fllwig hlds with C = 500: Cmpleteess: Fr all η > 0 if Sudess: Φ G(δ) η = pt(υ) C(η + ε) Φ G(δ) ε /C lg (/ε) = pt(υ) C

10 rf. xecute the SS-t-G reducti with parameters R = ad ε the iput graph G = (V, ). δ This yields a ique Games istace Υ alphabet size R. Apply the reducti frm partial ique Games t ique Games preseted i Secti 4 with parameter c = 40e lg( /ε), t get the ique Games istace Υ. Cmpleteess If Φ G(δ) η the there exists a subset S such that µ(s) [δ/0, δ] ad Φ(S) ε. By chice f R, we have µ(s) [, ]. Hece by Therem 3., there exists 0R R a /0e-partial strategy with value ( 40eη 4ε) fr Υ. Fially, by Therem 4. there exists a strategy fr the game Υ succeedig with prbability at least 40eη 4ε ε. Sudess Let β =. Suppse t, let us say pt(υ) C β. By Therem 4., if pt(υ) β the there exists a -partial strategy fr 3c Υ with value at least β. Applyig Therem 3., we get that there exists a set S with Φ(S) 5β ad» δ µ(s) 480e lg(/ε), 6δ. ε If µ(s) < δ the pad it with arbitrary vertices t cstruct a set S such that µ(s ) = δ. O the ther had, if µ(s) > δ, radmly subsample a subset S S f vlume δ. I either case, it is easy t check that Φ(S ) ( 5β)ε /C lg (/ε) ε /C lg (/ε). Ackwledgmets: We thak Baz Barak fr suggestig the ti f partial games, which helped t imprve the presetati f the results. We als thak Sajeev Arra, Vekatesa Guruswami, Oded Regev ad rasad Tetali fr isightful discussis. 6. RFRNCS [] N. Al. igevalues ad expaders. Cmbiatrica, 6():83 96, 986. [] N. Al ad V. D. Milma. λ, isperimetric iequalities fr graphs, ad superccetratrs. Jural f Cmbiatrial Thery. Series B, 38:73 88, 985. [3] S. Arra, R. Impagliazz, W. Matthews, ad D. Steurer. Imprved algrithms fr uique games via divide ad cquer. I lectric Cllquium Cmputatial Cmplexity CCCTR: TR0-04, 00. [4] S. Arra, S. Kht, A. Klla, D. Steurer, M. Tulsiai, ad N. K. Vishi. ique games expadig cstrait graphs are easy: exteded abstract. I STOC, pages 8. ACM, 008. [5] S. Arra, S. Ra, ad. Vazirai. xpader flws, gemetric embeddigs ad graph partitiig. I rceedigs f the thirty-sixth aual ACM Sympsium Thery f Cmputig (STOC-04), pages 3, Jue [6] Y. Auma ad Y. Rabai. A O(lg k) apprximate mi-cut max-flw therem ad apprximati algrithm. SIAM Jural Cmputig, 7():9 30, Feb [7] B. Barak, M. Hardt, I. Haviv, A. Ra, O. Regev, ad D. Steurer. Rudig parallel repetitis f uique games. I FOCS, pages I Cmputer Sciety, 008. [8] M. Charikar, K. Makarychev, ad Y. Makarychev. Itegrality gaps fr sherali-adams relaxatis. I STOC, pages ACM, 009. [9] J. Cheeger. A lwer bud smallest eigevalue f a laplacia. rblems i Aalysis (apers dedicated t Salm Bcher), pages 95 99, 970. [0] Feige ad Schechtma. O the ptimality f the radm hyperplae rudig techique fr MAX CT. RSA: Radm Structures Algrithms, 0, 00. []. Feige, G. Kidler, ad R. O Dell. derstadig parallel repetiti requires uderstadig fams. I I Cferece Cmputatial Cmplexity, pages 79 9, 007. [] K. Gergiu, A. Mage, T. itassi, ad I. Turlakis. Itegrality gaps f - () fr vertex cver SDs i the lvész-schrijver hierarchy. I FOCS, pages I Cmputer Sciety, 007. [3] S. Kht. O the pwer f uique -prver -rud games. I STOC, pages ACM, 00. [4] S. Kht, G. Kidler,. Mssel, ad R. O Dell. Optimal iapprximability results fr max-cut ad ther -variable csps? SIAM J. Cmput., 37():39 357, 007. [5] S. Kht ad O. Regev. Vertex cver might be hard t apprximate t withi -ε. J. Cmput. Syst. Sci., 74(3): , 008. [6] S. Kht ad N. K. Vishi. The uique games cjecture, itegrality gap fr cut prblems ad embeddability f egative type metrics it l. I FOCS, pages I Cmputer Sciety, 005. [7] F. T. Leight ad S. Ra. Multicmmdity max-flw mi-cut therems ad their use i desigig apprximati algrithms. J. ACM, 46(6):787 83, 999. [8] N. Liial,. Ld, ad Y. Rabivich. The gemetry f graphs ad sme f its algrithmic applicatis. Cmbiatrica, 5():5 45, 995. [9]. Raghavedra. Optimal algrithms ad iapprximability results fr every CS? I STOC, pages ACM, 008. [0]. Raghavedra ad D. Steurer. Hw t rud ay CS. I FOCS, pages I Cmputer Sciety, 009. []. Raghavedra ad D. Steurer. Itegrality gaps fr strg SD relaxatis f NIQ GAMS. I FOCS, pages I Cmputer Sciety, 009. []. Raghavedra, D. Steurer, ad. Tetali. Apprximatis fr the isperimetric ad spectral prfile f graphs ad fr restricted eigevalues f diagally-dmiat matrices. I STOC. ACM, 00. T Appear. [3] A. Ra. arallel repetiti i prjecti games ad a ccetrati bud. I STOC, pages 0. ACM, 008. [4] R. Raz. A cuterexample t strg parallel repetiti. I FOCS, pages I Cmputer Sciety, 008. [5] G. Scheebeck, L. Trevisa, ad M. Tulsiai. A liear rud lwer bud fr Lvász-Schrijver SD relaxatis f vertex cver. I I Cferece Cmputatial Cmplexity, pages 05 6, 007.