On the Hardness of Approximating Multicut and Sparsest-Cut [Extended Abstract]

Transcription

1 On the Hardness of Approxmatng Multcut and Sparsest-Cut [Extended Abstract] Shuch Chawla Robert Krauthgamer Rav Kumar Yuval Raban D. Svakumar Abstract We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF DELETION problems are NP-hard to approxmate wthn every constant factor, assumng the Unque Games Conjecture of Khot [STOC, 2002]. A quanttatvely stronger verson of the conjecture mples napproxmablty factor of Ω(log log n). 1. Introducton In the MULTICUT problem the nput s an undrected graph G = (V, E) on n = V vertces and k pars of vertces {s, t } k =1, called demand pars, and the goal s to fnd a mnmum-sze subset of the edges M E whose removal dsconnects all the demand pars,.e., n the subgraph (V, E \ M) every s s dsconnected from ts correspondng vertex t. In the weghted verson of ths problem, the nput also specfes a postve cost c(e) for each edge e E and the goal s to fnd a multcut M whose total cost c(m) = e M c(e) s mnmal. Ths problem s known to be APX-hard [12]. We prove that f a strong verson of the Unque Games Conjecture of Khot [19] s true, then MULTICUT s NP-hard to approxmate to wthn a factor of Ω(log log n). Under the orgnal verson of ths conjecture, our reducton shows that for every constant L > 0, t s NP-hard to approxmate MULTICUT to wthn factor L. Computer Scence Department, Carnege Mellon Unversty, Pttsburgh, PA 15213, USA. Emal: shuch@cs.cmu.edu. Ths work s supported n part by an IBM PhD Fellowshp, and by NSF grants CCR and IIS , and part of t was done whle the author was vstng IBM Almaden Research Center. IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120, USA. Emal: {rob,rav,sva}@almaden.bm.com. Computer Scence Department, Technon Israel Insttute of Technology, Hafa 32000, Israel. Part of ths work was done whle the author was on sabbatcal leave at Cornell Unversty, whle vstng IBM Almaden Research Center, and whle vstng the Insttute for Pure and Appled Mathematcs at UCLA. Research at the Technon supported n part by ISF grant number 52/03 and BSF grant number Emal: raban@cs.technon.ac.l Our methods also yeld smlar bounds for SPARSEST- CUT and MIN-2CNF DELETION. The SPARSEST-CUT problem has the same nput as MULTICUT, but the goal s to fnd a subset of the edges M E that mnmzes the rato of M (n the weghted verson, the total cost of M) to the number of demand pars that are dsconnected n (V, E \ M). 1 Snce SPARSEST-CUT s not known to be APX-hard, our result gves the frst ndcaton that ths problem mght be hard to approxmate. In the MIN-2CNF DELETION problem the nput s a weghted set of clauses on n varables, each clause of the form (x y), where x and y are lterals, and the goal s to fnd a Boolean assgnment to the varables mnmzng the total weght of unsatsfed clauses. 2 Our results also extend to the CORRELATION CLUSTERING problem [7, 10, 13, 14] of mnmzng dsagreements n a weghted graph, because the approxmablty of ths problem s known to be equvalent to that of MULTICUT n weghted graphs [10, 14] Known results on MULTICUT, SPARSEST-CUT, and MIN-2CNF DELETION MULTICUT and SPARSEST-CUT are fundamental combnatoral problems, wth connectons to multcommodty flow, edge expanson, and metrc embeddngs. Both problems can be approxmated to wthn an O(log k) factor through lnear programmng relaxatons [25, 16, 6, 26]. These bounds match the lower bounds on the ntegralty gaps up to constant factors [25, 16]. MIN-2CNF DELE- TION can also be approxmated to wthn an O(log n) factor, as mpled by the results of Klen et al. [23], who gve an approxmaton-preservng reducton from ths problem to MULTICUT. Recently, startng wth the groundbreakng O( log n)-approxmaton for the unform demands case [4], mproved approxmaton algorthms have 1 In general, the demand pars may have postve weghts (demands), but for our purpose of napproxmablty results, t clearly suffces to consder the more restrcted defnton above. Our hardness results for SPARSEST- CUT do not apply to the specal case of unform-demand, n whch every par of vertces forms a demand par. 2 Note that the constrants n MIN-2CNF DELETION are restrcted to equalty (and effectvely non-equalty) constrants. 1

2 been developed for the SPARSEST-CUT problem usng a semdefnte programmng relaxaton [4, 11, 2]. The best approxmaton factor currently known for general demands s O( log k log log k) [2]. The obvous modfcaton of the semdefnte program used for SPARSEST-CUT to solve MULTICUT was recently shown to have an ntegralty rato of Ω(log k) [1], whch matches, up to constant factors, the approxmaton factor and ntegralty gap of prevously analyzed lnear programmng relaxatons for ths problem. On the hardness sde, t s known that MULTICUT s APX-hard [12],.e., there exsts a constant c > 1, such that t s NP-hard to approxmate MULTICUT to wthn a factor smaller than c. It should be noted that ths hardness of approxmaton holds even for k = 3, and that the value of c s not specfed theren, but t s certanly much smaller than 2. The MIN-2CNF DELETION problem s also known to be APX-hard, as follows, e.g., from lnear equatons modulo 2 [17]. Assumng the Unque Games Conjecture, Khot [19, Theorem 3] essentally obtaned an arbtrarly large constantfactor hardness for MIN-2CNF DELETION, and ths mples, usng the aforementoned reducton of [23], a smlar hardness factor for MULTICUT. These results are not noted n [19], and are weaker than our results n several respects. Frst, our quanttatve bounds are better; thus f a stronger, yet almost as plausble, verson of ths conjecture s true, then our lower bound on the approxmaton factor mproves to L = Ω(log log n), compared wth the roughly Ω((log log n) 1/4 ) hardness that can be nferred from [19]; ths can be vewed as progress towards provng tght napproxmablty results for MULTICUT. Second, by qualtatvely strengthenng our MULTICUT result to a bcrtera verson of the problem, we extend our hardness results to the SPARSEST-CUT problem. It s unclear whether Khot s reducton smlarly leads to a hardness result for SPARSEST- CUT. Fnally, our proof s smpler (both the reducton and ts analyss), and makes drect connectons to cuts (n a hypercube), and thus may prove useful n further nvestgaton of such questons. For SPARSEST-CUT, no hardness of approxmaton result was prevously known. Independent of our work, Khot and Vshno [22] have recently used a dfferent constructon to show an arbtrarly large constant factor hardness for SPARSEST-CUT assumng the Unque Games Conjecture; ther hardness factor could, n prncple, be pushed to (log log n) c, for some constant c > 0, assumng a stronger quanttatve verson of the conjecture. Addtonally, they prove an ntegralty rato lower bound of Ω((log log n) c ), for some fxed c > 0, for the semdefnte program relaxatons used n the recent approxmaton algorthms for SPARSEST-CUT The Unque Games Conjecture Unque 2-prover game s the followng problem. The nput s a bpartte graph G Q = (Q, E Q ), where each sde p = 1, 2 contans n = Q /2 vertces denoted q p 1,, qp n, and represents n possble questons to prover p. In addton, the nput contans for each edge (q 1, q2 j ) E Q a nonnegatve weght w(q 1, q2 j ). These edges wll be called queston edges, to dstngush them from edges n the MULTI- CUT nstance. Each queston to a prover s assocated wth a set of d dstnct answers, denoted by [d] = {1,..., d}. The nput also contans, for every edge (q 1, q2 j ) E Q, a bjecton b j : [d] [d], whch maps every answer of queston q 1 to a dstnct answer for q2 j. A soluton A to the 2-prover game conssts of an answer A p [d] for each queston qp (.e., a sequence {Ap } over all p [2] and [n]). The soluton s sad to satsfy an edge (q 1, q2 j ) E Q f the answers A 1 and A2 j agree,.e., A 2 j = b j(a 1 ). We assume that the total weght of all the edges n E Q s 1 (by normalzaton). The value of a soluton s the total weght of all the edges satsfed by the soluton. The value of the game s the maxmum value achevable by any soluton to the game. Conjecture 1.1 (Unque Games [19]). For every fxed η, δ > 0 there exsts d = d(η, δ) such that t s NP-hard to determne whether a unque 2-prover game wth answer set sze d has value at least (1 η) or at most δ. We wll also consder stronger versons of the Unque Games Conjecture n whch η, δ, and d are functons of n. Specfcally, we wll consder versons wth max{η, δ} 1/(log n) Ω(1) and d = d(η, δ) O(log n). We denote the sze of an nput nstance by N. Notce that N = (n2 d ) Θ(1), and s thus polynomal n n as long as d O(log n), and n partcular for fxed d. Plausblty of the conjecture and ts stronger verson. The Unque Games Conjecture has been used to show optmal napproxmablty results for VERTEX COVER [21] and MAX-CUT [20, 27]. Provng the conjecture usng current technques appears qute hard. In partcular, the asserted NP-hardness s much stronger than what we can obtan va standard constructons usng the PCP theorem [5, 3] and the parallel repetton theorem [28], two deep results n computatonal complexty. Although the conjecture seems dffcult to prove n general, some specal cases are well-understood. In partcular, f at all the Unque Games Conjecture s true, then necessarly d max{1/η 1/10, 1/δ}. Ths follows from a semdefnte programmng algorthm presented n [19]. Our Ω(log log n) hardness result (see Corollary 1.4 below) requres the exstence of a constant c > 0, such that max{η, δ} 1/(log n) c and d O(log n), whch s not

3 excluded by the above. Fege and Rechman [15] recently showed that for every constant L > 0 there exsts a constant δ > 0, such that t s NP-hard to dstngush whether a unque 2-prover game (wth d = d(l, δ)) has value at least Lδ or at most δ; ths result falls short of the Unque Games Conjecture n that Lδ s bounded away from Our results We prove the followng hardness of approxmaton for MULTICUT, SPARSEST-CUT, and MIN-2CNF DELE- TION based on the Unque Games Conjecture. Theorem 1.2. Suppose that for η = η(n), δ = δ(n), and d = d(η, δ) O(log n), t s NP-hard to determne whether a unque 2-prover game wth Q = 2n vertces and answer set sze d has value at least 1 η(n) or at most δ(n). Then there exsts L(n) = Ω ( log 1 η(n Ω(1) )+δ(n Ω(1) ) ) such that t s NP-hard to approxmate MULTICUT, SPARSEST-CUT, and MIN-2CNF DELETION to wthn factor L(n). Ths theorem mmedately mples the followng two specfc hardness results. Corollary 1.3. The Unque Games Conjecture mples that, for every constant L > 0, t s NP-hard to approxmate MULTICUT, SPARSEST-CUT, and MIN-2CNF DELE- TION to wthn factor L. Corollary 1.4. The stronger verson of the Unque Games Conjecture n whch max{η, δ} 1/(log n) Ω(1), and d = d(η, δ) O(log n), mples that for some fxed c > 0, t s NP-hard to approxmate MULTICUT, SPARSEST-CUT, and MIN-2CNF DELETION to wthn factor c log log n. For SPARSEST-CUT our hardness results hold only for the search verson (n whch the algorthm needs to produce a cutset and not only ts value), snce our proof employs a Cook reducton Prelmnares Regular Unque Games. A unque 2-prover game s called regular f the total weght of queston edges ncdent at any sngle vertex s the same,.e., 1/n, for every vertex n Q. We now show that we can assume wthout loss of generalty that the graph n the Unque Games Conjecture s regular. For smplcty, we state ths only for fxed η and δ. A smlar result holds when they depend on n, because we ncrease the nput sze by no more than a polynomal factor, and ncrease η and δ by no more than a constant factor. Lemma 1.5. The Unque Games Conjecture mples that for every fxed η, δ > 0, there exsts d = d(η, δ) such that t s NP-hard to decde f a regular unque 2-prover game has value at least 1 η or at most δ. The proof s gven n Appendx A, and s based on an argument of Khot and Regev [21, Lemma 3.3]. The dependence of d on η and δ s mportant for our purposes. We thus pont out that ths argument does not change d = d(η, δ), and ncreases the sze of the nstance by at most a polynomal factor n n. Ths s acceptable n the settng of Theorem 1.2, snce the requrement d = d(η, δ) O(log n) s mantaned and only the unspecfed constants theren are affected. Bcrtera MULTICUT. Our proof for the hardness of approxmatng SPARSEST-CUT reles on a generalzaton of MULTICUT, where the soluton M s requred to cut only a certan fracton of the demand pars. For a gven graph G = (V, E), a subset of the edges M E wll be called a cutset of the graph. A cutset whose removal dsconnects all the demand pars s a multcut. An algorthm s called an (α, β)-bcrtera approxmaton for MULTICUT f, for every nput nstance, the algorthm outputs a cutset M that dsconnects at least α fracton of the demands and has cost at most β tmes that of the optmum multcut. In other words, f M s the least cost cutset that dsconnects all the k demand pars, then M dsconnects at least αk demand pars and c(m) β c(m ). Hypercubes, dmenson cuts, and antpodal vertces. As usual, the d-dmensonal hypercube (n short a d-cube) s the graph C = (V C, E C ) wth the vertex set V C = {0, 1} d, and an edge (u, v) E C for every two vertces u, v {0, 1} d that dffer n exactly one dmenson (coordnate). An edge (u, v) s called a dmenson-a edge, for a [d], f u and v dffer n dmenson a,.e., u v = 1 a where 1 a s a unt vector along dmenson a. The set of all the dmensona edges n the hypercube s called the dmenson-a cut n the hypercube; a dmenson cut s a dmenson-a cut for some dmenson a. The antpode of a vertex u s the (unque) vertex u all of whose coordnates are dfferent from those of u,.e., u = u 1 where 1 s the vector wth 1 n every coordnate. Notce that v s the antpode of u f and only f u s the antpode of v; thus, u, u form an antpodal par. The followng smple fact wll be key n our proof. Fact 1.6. In any hypercube, a sngle dmenson cut dsconnects every antpodal par. Organzaton. In Secton 2 we prove the part of Theorem 1.2 regardng the MULTICUT problem; our proof wll actually hold for bcrtera approxmaton for MULTICUT. We wll then show n Secton 3 that ths stronger result yelds a smlar hardness of approxmaton for SPARSEST-CUT. Fnally, n Secton 4, we modfy the reducton to obtan a hardness of approxmaton for MIN-2CNF DELETION.

4 2. Hardness of bcrtera approxmaton for MULTICUT In ths secton we prove the part of Theorem 1.2 regardng the MULTICUT problem, namely, that the Unque Games Conjecture mples that t s NP-hard to approxmate MULTICUT wthn a certan factor L. Our proof wll actually show a stronger result for every α 7/8 t s NP-hard to dstngush between whether there s a multcut of cost less than n2 d+1 (the YES nstance) or whether every cutset that dsconnects at least αk demand pars has cost at least n2 d+1 L (the NO nstance). Ths mples that t s NP-hard to obtan an (α, L)-bcrtera approxmaton for MULTICUT. We start by descrbng a reducton from unque 2-prover game to MULTICUT (Secton 2.1), and then proceed to analyze the YES nstance (Secton 2.2) and the NO nstance (Sectons 2.3 and 2.4). Fnally, we dscuss the gap that s created for a bcrtera approxmaton of MULTICUT (Secton 2.5) The reducton Gven a unque 2-prover game nstance G Q = (Q, E Q ) wth n = Q /2 and the correspondng edge weghts w(e) and bjectons b j : [d] [d], we construct a MULTICUT nstance G = (V, E) wth demand pars, as follows. For every vertex (.e., queston) q p Q, construct a d-dmensonal hypercube C p j ; the dmensons n ths cube correspond to answers for the queston q p j.3 For each of the 2n hypercubes, we let the edges nsde the hypercube have cost 1, and call them hypercube edges. For each queston edge (q 1, q2 j ) E Q, we extend b j (n the obvous way) to a bjecton from the vertces of C 1 to the vertces of Cj 2, and denote the resultng bjecton by b j : {0, 1}d {0, 1} d. Formally, for every u {0, 1} d (vertex n C 1 ) and every a [d], the a-th coordnate of b j (u) s gven by (b j (u)) a = u b 1 j (a). Then, we connect every vertex u C 1 to the correspondng vertex b j (u) Cj 2 usng an edge of cost w jλ, where Λ = n/η s a scalng factor. These edges are called cross edges. Denote the resultng graph by G = (V, E). Notce that V s smply the unon of the vertex sets of the hypercubes C p, for all p [2] and [n], and that the edge set E contans two types of edges, hypercube edges and cross edges. To complete the reducton, t remans to defne the demand pars. For a vertex u V, the antpode of u n G, denoted u, s defned to be the antpodal vertex of u n the hypercube C p that contans u. The set D of demand pars then 3 Ths s a standard technque n PCP constructons for graph optmzaton problems. A hypercube can be nterpreted as a long code [8], and a dmenson cut s the encodng of an answer n the 2-prover game. contans every par of antpodal vertces n G, and hence k = D = n2 d 1. Note that every vertex of G belong to exactly one demand par The YES nstance Lemma 2.1. If there s a soluton A for the unque 2-prover game G Q such that the total weght of the satsfed questons s at least 1 η, then there exsts a multcut M E for the MULTICUT nstance G such that c(m) 2 d+1 n. Proof. Let A be such a soluton for G Q. Construct M by takng the followng edges. For every queston q p Q and the correspondng answer A p (of prover p), take the dmenson-a p cut n cube Cp. In addton, for every edge (q 1, q2 j ) E Q that the soluton A does not satsfy, take all the cross edges between the correspondng cubes C 1 and Cj 2 Ẇe frst clam that removng M from G dsconnects all the demand pars. To see ths, we defne a Boolean functon f : V {0, 1} on the graph vertces. For every cube C p, consder the dmenson-ap cut; t dsconnects the cube nto two connected components, one contanng the all zeros vector 0 and one contanng the all ones vector 1. For every v C p, let f(v) = 0 f v s n the same sde as 0, and f(v) = 1 otherwse. Ths s exactly the A p -th bt n v,.e., f(v) = v A p. Now consder any demand par (v, v), and note that f(v) = 1 f(v). We wll show below that every edge (u, v) / M satsfes the property f(u) = f(v). Ths clearly proves the clam. Consder frst a hypercube edge (u, v) n C p that s not a dmenson-a p edge. Then f(u) = u A p = v A p = f(v), by the defnton of f. Next consder a cross edge (u, v) / M. Then ths edge les between cubes C 1 and Cj 2, such that the queston edge (q 1, q2 j ) satsfed by the unque 2-prover game soluton A. Therefore, b j (A 1 ) = A2 j. Then, f(u) = u A 1 = v bj (A 1 ) = v A 2 j = f(v). Fnally, we bound the cost of the soluton. Let S be the set of queston edges not satsfed by the soluton A. The total cost of the multcut soluton s thus c(m) = 2n 2 d d Λ (Q 1,Q2 j ) S w j 2 d n + 2 d n η η = 2d+1 n Hypercube cuts and nfluences We wll analyze the NO nstance shortly, but frst we set up some notaton and present a few techncal lemmas regardng cuts n hypercubes. In partcular, we present Lemma 2.3, whch wll have a crucal role n that analyss. Recall that the dmensons of the hypercubes n the multcut nstance correspond to answers to the 2-prover game. Therefore, we defne the extent to whch a dmenson partcpates n a cut on the cube as follows. Let C = (V C, E C ) be a d-dmensonal hypercube. For a functon f : V C R,

5 the nfluence of dmenson a [d] (a.k.a. the nfluence of the a-th varable) on the functon, denoted Ia f, s defned to be the fracton of dmenson-a edges (u, v) E C for whch f(u) f(v). For a cutset M E C, the nfluence of dmenson a [d] on the cutset, denoted Ia M, s defned as the fracton of dmenson-a edges that belong to M. Observe that M = 2 d 1 a [d] IM a. Proposton 2.2. Let M E C be a cutset n a hypercube C = (V C, E C ). Defne g : V C Z by labelng the connected components of C \M by dstnct ntegers, and lettng g(v) for v V C be the label of the connected component contanng v. Then I M a I g a. Proof. Observe that the cutset M must contan every edge (u, v) E C for whch g(u) g(v). The lemma below shows that f a cutset M has few edges (.e., small cost) but ts removal dsconnects a large fracton of the antpodal pars n the hypercube C, then there must be a dmenson a [d] wth large nfluence. Lemma 2.3. Let M be a cutset n a d-dmensonal hypercube C, and suppose that removng M dsconnects at least β fracton of the antpodal pars n C. Then for all x > 0, Ia M βx max a [d] IM a 2 6x /27. a [d] To prove ths, we wll make use of the followng lemma, due to Kahn, Kala, and Lnal [18] (see also [29, Secton 1.5]). Lemma 2.4 (Kahn, Kala, and Lnal [18]). Let f be a Boolean functon defned on a hypercube, and suppose the fracton of nputs x for whch f(x) = 1 s p 1/2. Then for all α > 0, 1 α I f + (I f )4/3 2p log α α. We note that the proof of Lemma 2.4 s based on Fourer analyss of Boolean functons, and that ts statement above follows from the proofs theren. Proof of Lemma 2.3. We frst convert the cutset M nto a two-sded (bnary) cut. Observe that each connected component of C \ M must have sze at most 2 d β2 d 1 = (1 β/2) V C. If there s a component of sze larger than V C /2, we combne the rest of the components nto a sngle component. Otherwse, we splt the set of components nto two parts such that the total sze of the components n each part s at most 2 3 V C. Call the resultng cutset M. Note that M M and thus, for every a [d], the nfluence of every dmenson n M s no larger than ts counterpart n M,.e., I M a Ia M. Hence, a IM a a IM a βx. Ths twosded cut defnes a Boolean functon f : V C {0, 1} wth balance p 1/2 satsfyng p mn{β/2, 1/3} β/3 and I M a = I f. Usng Lemma 2.4 wth α = 22x, we have We thus obtan βx 2 2x + (I M a ) 4/3 2 β 2x 3 2 2x a [d] (I M a ) 4/3 β x 3 2 2x a Now set y = max a [d] I M a. Then we get a (I M a ) 4/3 y 1/3 a I M a βxy 1/3 Therefore, we have y 1/ x, or, y 2 6x /27. The next lemma shows that f two functons f, g : V C R agree on most of the nputs v V C, then ther nfluences are qute smlar. Lemma 2.5. Let C = (V C, E C ) be a hypercube. If for two functons f, g : V C R we have f(v) = g(v) for all but a γ fracton of nputs v V C, then for every dmenson a we have I f a I g a 2γ. Proof. Suppose that C s a d-dmensonal hypercube, and consder a dmenson-a edge (u, v) E C. By our assumpton, for all but at most γ2 d such edges, we must have f(u) = g(u) and f(v) = g(v), and n partcular f(u) f(v) = g(u) g(v). Recallng that there are exactly 2 d 1 dmenson-a edges, and that Ia f s the fracton of those edges for whch f(u) f(v) 0 (and smlarly for g), we conclude that f(u) f(v) = g(u) g(v) for at most 2γ fracton of the dmenson-a edges, and thus Ia f Ia g 2γ The NO nstance Lemma 2.6. There exsts L = Ω(log 1/(η + δ)) such that f the MULTICUT nstance G has a cutset of cost at most 2 d nl whose removal dsconnects α 7/8 fracton of the demand pars, then there s a soluton A for the unque 2- prover game G Q whose value s larger than δ. Proof. Let L = c log 1/(η + δ) where c > 0 s a constant to be determned later, and let M E be a cutset of cost c(m) 2 d nl whose removal dsconnects α 7/8 fracton of the demand pars. Usng M, we wll construct for the unque 2-prover game G Q a randomzed soluton A whose expected value s larger than δ, thereby provng the exstence of a soluton of value larger than δ. Wthout loss of generalty, we may assume that M s mnmal wth respect to contanment, namely, for every subset M M, f M M then removng M from G dsconnects fewer demand pars than removng M would. Gven such a mnmal

6 cutset M, for each cube C p n G, consder the cutset M nduces n ths cube, and let Ia p, be the nfluence of dmenson a [d] on ths cutset. The randomzed soluton A (.e., a strategy for the two provers) s defned as follows. For each vertex (queston) q p Q, we choose A p to be the answer (dmenson) a [d] wth probablty I p, a / a [d] Ip, a. We proceed to analyze the expected value of ths randomzed soluton A. Recall that the value of a soluton corresponds to the probablty that, for a queston edge (q 1, q2 j ) chosen at random wth probablty proportonal to ts weght, we have a 2 j = b j(a 1 ). Notce that although q1 and qj 2 are correlated, each one s unformly dstrbuted because Q s regular. Wthout loss of generalty, we assume removng M dsconnects at least as many demand pars nsde the cubes {Cl 1} l [n] as nsde the cubes {Cl 2} l [n]. Now we clam that wth a hgh probablty over the choce of a queston edge, the cut M has a low cost over edges ncdent on the correspondng hypercubes, and dsconnects many demand pars n the hypercubes. In other words, the qualty of the cut locally s nearly as good as the qualty of the cut globally. In partcular, we upper bound the probablty of the followng four bad events (for a choce of a queston edge (q 1, q2 j )): E 1 = fewer than half the demand pars n C 1 are dsconnected n G \ M. E 2 = M contans more than 2 d+2 L hypercube edges n C 1. E 3 = M contans more than 2 d+2 L hypercube edges n C 2 j. E 4 = M contans more than 2 d /2 96L+7 cross edges between C 1 and C2 j. Frst, by our assumpton above, removng M dsconnects at least α 7/8 fracton of the demand pars nsde the cubes {Cl 1} l [n], and thus by Markov s nequalty, Pr[E 1 ] 1/4. Next, the cutset M contans at most 2 d nl hypercube edges, thus the expected number of edges n C 1 C2 j that are contaned n M s at most 2 d L, and Pr[E 2 E 3 ] 1/2. Fnally, f c > 0 s suffcently small, Pr[E 4 ] ηl2 96L+7 η 1/2 1/8, as otherwse the total cost along the correspondng queston-edges (q 1, q2 j ) (.e., those for whch the cutset M contans more than 2 d /2 96L+7 cross edges between C 1 and Cj 2) s more than (ηl296l+7 ) (2 d /2 96L+7 ) (n/η) = n2 d L c(m). Takng a unon bound, we upper bound the probablty that any of the bad events occurs by Pr[E 1 E 2 E 3 E 4 ] 7 8. In order to lower bound the expected value of the randomzed soluton A, we would lke to show that f none of the above bad events happens, then there exsts a dmenson a [d], such that n cube C 1 ths dmenson a has large nfluence, I 1, a, and n cube C2 j dmenson b j(a ) has large nfluence, I 2,j b j (a ). For the cube C1, f the event E 2 does not occur, then a [d] I1, a 8L. If nether E 1 nor E 2 occurs, then we can use Lemma 2.3 (wth β = 1/2, x = 16L) and conclude that there exsts a dmenson a [d] such that I 1, a 2 96L /27. For the sake of analyss, label the connected components of G \ M wth dstnct nteger values. Defne f : C 1 Z by lettng f(v) for u C 1 be the label of the connected component of u, and defne g : Cj 2 Z smlarly. For every u C 1, f f(u) g(b j (u)) then the cross edge (u, b j (u)) must be contaned n the cutset M, and because we assumed the event E 4 happens, ths occurs for at most 2 d /2 96L+7 vertces u C 1. Furthermore, by the defnton of f and g, we have Ia 1, = Ia f and Ia 2,j = Ia. g Applyng Lemma 2.5 to the functons f and g b j, we conclude that Ia f I g b j a 2 96L 6 for all dmensons a [d]. Fnally, snce b j s just a permutaton of the coordnates, for all we have a [d], I g b j a and thus = I g b j (a). Altogether, we obtan I 2,j b j (a ) I1, a 2 96L L /54, Pr[A 2 j = b j (A 1 )] Pr[A 1 = a, A 2 j = b j (a )] 1 8 I 1, a a [d] I1, a Ω(L L ) I 2,j a a [d] I2,j b j(a) We conclude that the expected value of the randomzed soluton A s (,j) E Q w(q 1, q 2 j ) Pr[A 2 j = b j (A 1 )] Ω(L L ) > δ, where the last nequalty holds f c > 0 s suffcently small, and ths completes the proof of Lemma Puttng t all together The above reducton from unque 2-prover game to MULTICUT produces a gap of L(n) = Ω(log 1/(η(n) + δ(n))). We assumed d(η, δ) O(log n), and thus the resultng MULTICUT nstance G has sze N = (n2 d ) O(1) = n Θ(1). It follows that n terms of the nstance sze N, the gap s L(N) = Ω(log 1/(η(N Θ(1) ) + δ(n Θ(1) ))). Ths completes the proof of the part of Theorem 1.2 regardng the MULTICUT problem, namely, that the Unque Games Conjecture mples that t s NP-hard to approxmate MULTICUT wthn the above factor L(N). In fact, the above proof shows that t s even NP-hard to obtan a (7/8, L(N))- bcrtera approxmaton.

7 Note that the number of demand pars s k = n2 d 1 = n Θ(1), and thus the hardness of approxmaton factor s smlar when expressed n terms of k as well. Note also that all edge weghts n the MULTICUT nstance constructed above are bounded by a polynomal n the sze of the graph. Therefore, va a standard reducton, a smlar hardness result holds for the unweghted MULTICUT problem as well. 3. Hardness of approxmatng SPARSEST-CUT In ths secton we prove the part of Theorem 1.2 regardng the Sparsest-Cut problem. The proof follows mmedately from the next lemma n conjuncton wth the hardness of bcrtera approxmaton of MULTICUT (from the prevous secton). Lemma 3.1. Let 0 < α < 1 be a constant. If there exsts a polynomal-tme algorthm for SPARSEST-CUT that produces a cut whose value s wthn factor 1 of the mnmum, then there s a polynomal tme algorthm that computes an (α, TICUT. 2 1 α )-bcrtera approxmaton for MUL- Proof. Fx 0 < α < 1, and suppose A s a polynomal-tme algorthm for SPARSEST-CUT that produces a cut whose value s wthn factor 1 of the mnmum. Now suppose we are gven an nput graph G = (V, E) and k demand pars {s, t } k =1. We may assume wthout loss of generalty that every s s connected (n G) to ts correspondng t. Let c mn and c max be the smallest and largest edge costs n G, and let n = V. We now descrbe the bcrtera approxmaton algorthm for MULTICUT. For every value C [c mn, n 2 c max ] that s a power of 2, execute a procedure that we wll descrbe momentarly to compute a cutset M C E, and report, from all these cutsets M C whose removal dsconnects at least αk demand pars, the one of least cost. For a gven value C > 0, the procedure starts wth M C =, and then teratvely augments M C as follows: Take a connected component S of G \ M C, apply algorthm A to G[S] (the subgraph nduced on S and all the demand pars that le nsde S), and f the resultng cutset E S has value (n G[S]) 1 α C at most k, then add the edges E S to M C. Here, the value (rato of cost to demands cut) of E S s defned as b S = c(e S )/ D S, where D S s the collecton of demand pars that le n G[S] and get dsconnected (n G[S]) when E S s removed. Proceed wth the teratons untl for every 1 α C k, connected component S n G \ M C we have b S > at whch pont the procedure returns the cutset M C. Ths algorthm clearly runs n polynomal tme. To analyze ts performance, we frst clam that for every value C, the cutset M C returned by the above procedure has sparsestcut value (rato of cost to demand dsconnected, n G) at C most 1 α k. Indeed, suppose the procedure performs t augmentaton teratons. Denote by S the connected component S that s cut at teraton [t], by E S the correspondng cutset output by A, and by D S the correspondng set of demand pars that get dsconnected. Clearly, M C s the dsjont unon E 1 E t, and t s easy to verfy that the collecton D C of demand pars cut by the cutset M C s the dsjont unon D S1 D St. Thus, c(m C ) = t c(e S ) =1 = 1 α C k t D S =1 1 α C k D C, whch proves the clam. For the sake of analyss, fx an optmal multcut M E,.e., a cutset of G whose removal dsconnects all the demand pars and has the least cost. The sparsest-cut value of M s b = c(m )/k. We wll show that f C [c(m ), 2c(M )], then the above procedure produces a cutset M C whose removal dsconnects a collecton D C contanng D C αk demand pars; ths wll complete the proof of the lemma, because t mmedately follows that c(m C ) 1 α C k D C 1 α 2c(M ), and clearly c(m ) [c mn, ( n 2) cmax ]. So suppose now C [c(m ), 2c(M )] and assume for contradcton that D C < αk. Denote by V 1,..., V p V the connected components of G \ M C, and let D j contan the demand pars that le nsde V j. It s easy to see that p j=1 D j = k D C > (1 α)k. Smlarly, let Mj be the collecton of edges n M that le nsde V j. Then c(m ) p j=1 c(m j ). Notce that, n every nduced graph G[V j], the edges of Mj form a cutset (of G[V j]) that cuts all the demand pars n D j. Usng the stoppng condton of the procedure, and snce A provdes an approxmaton wthn factor, we have c(mj ) 1 C 1 α k D j (the nequalty s not strct because D j mght be empty). We thus derve the contradcton p c(m ) c(mj ) 1 1 α C p D j > c(m ). k j=1 j=1 Ths shows that when C [c(m ), 2c(M )], the procedure stops wth a cutset M C whose removal dsconnects D C αk demand pars, and concludes the proof of the lemma. 4. Hardness of approxmatng MIN-2CNF DELETION In ths secton, we modfy the reducton n Secton 2.1 to obtan a hardness of approxmaton for MIN-2CNF

8 DELETION. In partcular, we reduce the MULTICUT nstance obtaned n Secton 2.1 to MIN-2CNF DELETION, such that a soluton to the latter gves a MULTICUT of the same cost n the former. The MIN-2CNF DELETION nstance contans 2 d 1 n varables, one for each demand par (u, u). In partcular, for every demand par (u, u) D, we assocate the lteral x u wth u and the lteral x u = x u wth u. For every edge e = (u, v) n the graph G there s a clause (x u x v ) whose weght s equal to the edge-weght w e. The followng lemma s mmedate from the constructon and mples an analog of Lemma 2.6 for MIN-2CNF DELETION. Lemma 4.1. Gven an assgnment S of cost W to the above nstance of MIN-2CNF DELETION, we can construct a soluton of cost W to the MULTICUT nstance G. Proof. Let M be the set of edges (u, v) for whch S(x u ) S(x v ). Then M corresponds to the clauses that are not satsfed by S and has weght W. The lemma follows from observng that M s ndeed a multcut S s constant over connected components n G \ M, and for any demand par (u, u), S(x u ) S(x u ). We now gve an analog of Lemma 2.1. Lemma 4.2. If there s a soluton A for the unque 2-prover game G Q such that the total weght of the satsfed questons s at least 1 η, then there exsts an assgnment S for the above MIN-2CNF DELETION nstance such that c(s) 2 d+1 n. Proof. Gven the soluton A for G Q, we construct an assgnment S as follows. For every queston q p and for every vertex u n the correspondng hypercube C p, defne S(x u) to be the A p -th bt of u,.e., S(x u) = u A p. Note that ths s a vald assgnment,.e., S(x u ) = 1 S(x u ) for all vertces u, as u A p = 1 u A p. We bound the cost of the soluton by frst analyzng the clauses correspondng to hypercube edges n the correspondng MULTICUT nstance. Consder unsatsfed clauses contanng both varables n the same hypercube C p, and note that the hypercube edges correspondng to these clauses form a dmenson-a p cut n the cube Cp. Therefore, the total weght of these clauses s at most (2 d 1 )(2n) = 2 d n. Fnally, consder an unsatsfed clause (x u x v ) correspondng to vertces n dfferent hypercubes C 1 and C2 j. Then S(x u ) S(x v ) mples that u A 1 = v bj (A 1 ) v A 2 j, or, b j (A 1 ) A2 j. There are at most 2d such clauses for each queston par not satsfed by the soluton A. Therefore, the total weght of such clauses s at most 2 d n η η = 2 d n. The lemma follows from addng the two costs. Lemmas 4.1 and 4.2 along wth Lemma 2.6 mply the part of Theorem 1.2 regardng MIN-2CNF DELETION. 5. Concludng remarks Several mportant questons are left open. Frst, one would lke to elmnate the dependence on the Unque Games Conjecture, and obtan a standard hardness of approxmaton result. Yet another challenge s to mprove the hardness factor. For MULTICUT, the Ω(log n) ntegralty rato lower bound of [1] suggests that the napproxmablty bound may be mproved. In partcular, (log n) c hardness for a constant c > 1/2 wll separate the approxmablty of MULTICUT from that of SPARSEST-CUT (n lght of the recent approxmaton due to [2]). The man bottleneck to mprovng the hardness factor les n Lemma 2.3, whch n turn crucally depends on Lemma 2.4, due to [18]. These bounds are tght n general, as shown by the trbes functon [9]. However, n our context, n the reducton to the (non-bcrtera) MULTICUT problem, one may addtonally assume that f s odd, that s, f(u) f(u) for all nputs u (because a multcut should separate all the antpodal demand pars). Even wth ths addtonal assumpton, our Ω(log log n) bound cannot be mproved substantally, as demonstrated by the followng varant of the trbes functon [24]: Partton the varables u 1,..., u d nto subsets of sze log d 2 log log d each; the output s the value of the frst unanmous subset (under an arbtrary orderng), or u 1 f no unanmous trbe exsts. Ths functon s clearly odd, yet all varables have nfluence at most O( log2 d d ) and the total nfluence s O(log d). For Lemma 2.3, ths functon leads to a cutset M wth β = 1, such that for x = a IM a = O(log d) we have max a Ia M 2 Ω(x). A thrd challenge s to obtan hardness of approxmaton results for the unform-demand case of the SPARSEST-CUT problem or for the BALANCED-CUT problem. Our results do not apply to ths specal but mportant case; n partcular, f a 2-prover system has a low-cost balanced cut, then the correspondng graph on hypercubes would have a low-cost balanced cut regardless of the value of the 2-prover game. Alternatvely, of course, one mght mprove the approxmaton algorthms for any of these problems. References [1] A. Agarwal, M. Charkar, K. Makarychev, and Y. Makarychev. O( log n) approxmaton algorthms for Mn UnCut, Mn 2CNF Deleton, and drected cut problems. In Proceedngs of the 37th Annual ACM Symposum on Theory of Computng, 2005.

9 [2] S. Arora, J. R. Lee, and A. Naor. Eucldean dstorton and the sparsest cut. In Proceedngs of the 37th Annual ACM Symposum on Theory of Computng, [3] S. Arora, C. Lund, R. Motwan, M. Sudan, and M. Szegedy. Proof verfcaton and the hardness of approxmaton problems. Journal of the ACM, 45(3): , [4] S. Arora, S. Rao, and U. Vazran. Expander flows, geometrc embeddngs, and graph parttonngs. In Proceedngs of the 36th Annual ACM Symposum on Theory of Computng, pages , [5] S. Arora and S. Safra. Probablstc checkng of proofs: A new characterzaton of NP. Journal of the ACM, 45(1):70 122, [6] Y. Aumann and Y. Raban. An O(log k) approxmate mncut max-flow theorem and approxmaton algorthm. SIAM Journal on Computng, 27(1): , [7] N. Bansal, A. Blum, and S. Chawla. Correlaton Clusterng. Machne Learnng, Specal Issue on Clusterng, 56(1-3):89 113, [8] M. Bellare, O. Goldrech, and M. Sudan. Free bts, PCP s and non-approxmablty towards tght results. SIAM Journal on Computng, 27(3): , [9] M. Ben-Or and N. Lnal. Collectve con flppng. In Randomness and Computaton, pages , [10] M. Charkar, V. Guruswam, and A. Wrth. Clusterng wth qualtatve nformaton. In Proceedngs of the 44th IEEE Symposum on Foundatons of Computer Scence, pages , [11] S. Chawla, A. Gupta, and H. Räcke. Improved approxmatons to sparsest cut. In Proceedngs of the 16th Annual ACM-SIAM Symposum on Dscrete Algorthms, pages , [12] E. Dahlhaus, D. S. Johnson, C. H. Papadmtrou, P. D. Seymour, and M. Yannakaks. The complexty of multtermnal cuts. SIAM Journal on Computng, 23(4): , [13] E. Demane and N. Immorlca. Correlaton clusterng wth partal nformaton. In Proceedngs of the 6th Internatonal Workshop on Approxmaton Algorthms for Combnatoral Optmzaton Problems (APPROX), pages 1 13, [14] D. Emanuel and A. Fat. Correlaton clusterng mnmzng dsagreements on arbtrary weghted graphs. In Proceedngs of 11th Annual European Symposum on Algorthms, pages , [15] U. Fege and D. Rechman. On systems of lnear equatons wth two varables per equaton. In Proceedngs of the 7th Internatonal Workshop on Approxmaton Algorthms for Combnatoral Optmzaton Problems (AP- PROX), pages , [16] N. Garg, V. V. Vazran, and M. Yannakaks. Approxmate max-flow mn-(mult)cut theorems and ther applcatons. SIAM Journal on Computng, 25(2): , [17] J. Håstad. Some optmal napproxmablty results. Journal of the ACM, 48(4): , [18] J. Kahn, G. Kala, and N. Lnal. The nfluence of varables on boolean functons. In Proceedngs of the 29th IEEE Symposum on Foundatons of Computer Scence, pages 68 80, [19] S. Khot. On the power of unque 2-prover 1-round games. In Proceedngs of the 34th Annual ACM Symposum on Theory of Computng, pages , [20] S. Khot, G. Kndler, E. Mossel, and R. O Donnell. Optmal napproxmablty results for MAX-CUT and other 2- varable CSPs. In Proceedngs of the 45th IEEE Symposum on Foundatons of Computer Scence, pages , [21] S. Khot and O. Regev. Vertex cover mght be hard to approxmate to wthn 2-ɛ. In Proceedngs of the 18th Annual IEEE Conference on Computatonal Complexty, pages , [22] S. Khot and N. Vshno. On embeddablty of negatve type metrcs nto l 1. Manuscrpt, [23] P. Klen, A. Agrawal, R. Rav, and S. Rao. Approxmaton through multcommodty flow. In Proceedngs of the 31st IEEE Symposum on Foundatons of Computer Scence, pages , [24] R. Kumar, D. Svakumar, and E. Vee. Manuscrpt, [25] F. T. Leghton and S. Rao. An approxmate max-flow mncut theorem for unform multcommodty flow problems wth applcatons to approxmaton algorthms. In Proceedngs of the 29th IEEE Symposum on Foundatons of Computer Scence, pages , [26] N. Lnal, E. London, and Y. Rabnovch. The geometry of graphs and some of ts algorthmc applcatons. Combnatorca, 15(2): , [27] E. Mossel, R. O Donnell, and K. Oleszkewcz. Nose stablty of functons wth low nfluences: Invarance and optmalty. Manuscrpt, [28] R. Raz. A parallel repetton theorem. SIAM Journal on Computng, 27(3): , [29] D. Stefankovc. Fourer Transforms n Computer Scence. PhD thess, Unversty of Chcago, A. Regularty of the Unque Games nstance Proof of Lemma 1.5. Gven a unque 2-prover game Q, we descrbe how to convert t to a regular game whle preservng ts completeness and soundness. Frst we clam that we can assume that the rato between the maxmum weght max e w e and the mnmum weght mn e w e s bounded by n 3. Ths s because we can remove all edges wth weght less than 1 n max 3 e w e from the graph, changng the soundness and completeness parameters by at most 1 n. By a smlar argument, we can assume that all weghts n the graph are ntegral multples of t = 1 n mn 2 e w e. Now we convert Q to a regular graph Q as follows. For each prover p {1, 2} and queston q p, form W (p, )/t

10 vertces q p (1),, qp (W (p, )/t), where W (p, ) s the total weght of all the edges ncdent on q p. For every par of vertces (q 1, q2 j ), connected by an edge e n Q, we form an edge between q 1(x) and q2 j (y), for all possble values of x t t and y, wth weght w e W (1,) W (2,j). Note that the total weght of all the edges remans the same as before. Each new vertex q 1 (x) has total weght e w t t W (2,j) e W (1,) W (2,j) t = t, where the sum s over all edges e ncdent on q 1. Therefore, the graph s regular. Furthermore, the number of vertces ncreases by a factor of at most n 5. It only remans to show that the soundness and completeness parameters are preserved. To see ths, note that any soluton on the orgnal graph Q can be transformed to a soluton of the same value on Q, by pckng the same answer for every vertex q p (x) n Q as the answer pcked for q p n Q. Lkewse, consder a soluton n Q. Note that the answers for the questons q p (x) wth dfferent values of x must all be the same, because all these questons are connected to dentcal sets of vertces, wth the same weghts. Therefore, the soluton n Q that pcks the same answer for q p as the answer for qp (x) n Q has the same weght as the gven soluton n Q. Thus for every soluton n Q, there s a soluton of the same weght n Q and vce versa. Ths proves that the two games have exactly the same soundness and completeness parameters.