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34
Graph Expansion and the Unique Games Conjecture
, 2010
"... The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a dregular graph, the edge expansion/conductance Φ(S) of a subset E(S,V \S) dS S ⊆ V is defined as Φ(S) =. Approximating the conductance of small linear sized sets (size δn) is a natur ..."
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Cited by 41 (5 self)
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The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a dregular graph, the edge expansion/conductance Φ(S) of a subset E(S,V \S) dS S ⊆ V is defined as Φ(S) =. Approximating the conductance of small linear sized sets (size δn) is a natural optimization question that is a variant of the wellstudied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1 − ε), and close to 0 expansion. In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following: –We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs. This is the first nontrivial “reverse ” reduction from a natural optimization problem to Unique Games. –Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UGhard to approximate small set expansion. –On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].
Integrality gaps for strong SDP relaxations of unique games
"... Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner pro ..."
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Cited by 38 (7 self)
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Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp(Ω(log log n) 1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program by R rounds of Sherali–Adams liftandproject, we prove a Unique Games integrality gap for R = Ω(log log n) 1/4. By composing these SDP gaps with UGChardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGCbased hardness is known. Consequently, this work implies that including any valid constraints on up to exp(Ω(log log n) 1/4) vectors to natural semidefinite program, does not improve the approximation ratio for any problem in the following classes: constraint satisfaction problems, ordering constraint satisfaction problems and metric labeling problems over constantsize metrics. We obtain similar SDP integrality gaps for Balanced Separator, building on [11]. We also exhibit, for explicit constants γ, δ> 0, an npoint negativetype metric which requires distortion Ω(log log n) γ to embed into ℓ1, although all its subsets of size exp(Ω(log log n) δ) embed isometrically into ℓ1. Keywordssemidefinite programming, approximation algorithms, unique games conjecture, hardness of approximation, SDP hierarchies, Sherali–Adams hierarchy, integrality gap construction 1.
Unique games with entangled provers are easy
 In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
, 2008
"... We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only a ..."
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Cited by 32 (9 self)
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We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel ‘quantum rounding technique’, showing how to take a solution to an SDP and transform it to a strategy for entangled provers. Using our approximation by a semidefinite program we also show a parallel repetition theorem for unique entangled games. 1
Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives
 In Proceedings of 52nd Annual Symposium on Foundations of Computer Science (FOCS
, 2011
"... We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and ..."
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Cited by 25 (3 self)
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We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NPhardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for lowrank approximations of a matrix in Frobenius norm using columns of the matrix. For all the above graph problems, we give an algorithm running in time nO(r/ε 2) with approximation ratio 1+εmin{1,λr} , where λr is the r’th smallest eigenvalue of the normalized graph Laplacian L. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lowerorder terms of the stipulated bound. Our results imply (1 + O(ε)) factor approximation in time nO(r
INAPPROXIMABILITY RESULTS FOR MAXIMUM EDGE BICLIQUE, MINIMUM LINEAR ARRANGEMENT, AND SPARSEST CUT
, 2011
"... We consider the Minimum Linear Arrangement problem and the (Uniform) Sparsest Cut problem. So far, these two notorious NPhard graph problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme, unless NPcomplete problems ca ..."
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Cited by 21 (0 self)
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We consider the Minimum Linear Arrangement problem and the (Uniform) Sparsest Cut problem. So far, these two notorious NPhard graph problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme, unless NPcomplete problems can be solved in randomized subexponential time. Furthermore, we show that the same techniques can be used for the Maximum Edge Biclique problem, for which we obtain a hardness factor similar to previous results but under a more standard assumption.
Parallel Repetition of Entangled Games ∗
"... We consider oneround games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of provers? Classically, th ..."
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Cited by 16 (3 self)
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We consider oneround games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of provers? Classically, this question, open for many years, has culminated in Raz’s celebrated parallel repetition theorem on one hand, and in efficient product testers for PCPs on the other. In the case where provers share entanglement, the only previously known results are for special cases of games, and are based on techniques that seem inherently limited. Here we show for the first time that the maximum success probability of entangled games can be reduced through parallel repetition, provided it was not initially 1. Our proof is inspired by a seminal result of Feige and Kilian in the context of classical twoprover oneround interactive proofs. One of the main components in our proof is an orthogonalization lemma for operators, which might be of independent interest. Twoprover games play a major role both in theoretical computer science, where they led to many breakthroughs such as the discovery of tight inapproximability results, and in quantum physics, where they first arose in the context of Bell inequalities. In such games, a referee chooses a pair of questions
Randomly supported independence and resistance
 In 38th Annual ACM Symposium on Theory of Computation
, 2009
"... Abstract. We prove that for any positive integers q and k, there is a constant cq,k such that a uniformly random set of cq,knk log n vectors in [q] n with high probability supports a balanced kwise independent distribution. In the case of k ≤ 2 a more elaborate argument gives the stronger bound cq, ..."
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Cited by 14 (4 self)
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Abstract. We prove that for any positive integers q and k, there is a constant cq,k such that a uniformly random set of cq,knk log n vectors in [q] n with high probability supports a balanced kwise independent distribution. In the case of k ≤ 2 a more elaborate argument gives the stronger bound cq,knk. Using a recent result by Austrin and Mossel this shows that a predicate on t bits, chosen at random among predicates accepting cq,2t2 input vectors, is, assuming the Unique Games Conjecture, likely to be approximation resistant. These results are close to tight: we show that there are other constants, c ′ q,k, such that a randomly selected set of cardinality c ′ q,knk points is unlikely to support a balanced kwise independent distribution and, for some c> 0, a random predicate accepting ct2 / log t input vectors is nontrivially approximable with high probability. In a different application of the result of Austrin and Mossel we prove that, again assuming the Unique Games Conjecture, any predicate on t Boolean inputs accepting at least (32/33) · 2t inputs is approximation resistant. The results extend from balanced distributions to arbitrary product distributions.
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 12 (1 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
No strong parallel repetition with entangled and nonsignaling provers
 In CCC ’10: Proc. 25th Annu. IEEE Conf. on Computational Complexity
, 2010
"... ar ..."
Strong Parallel Repetition Theorem for Free Projection Games
, 2009
"... The parallel repetition theorem states that for any two provers one round game with value at most 1 − ɛ (for ɛ < 1/2), the value of the game repeated n times in parallel is at most (1−ɛ 3) Ω(n / log s) where s is the size of the answers set [Raz98],[Hol07]. For Projection Games the bound on the v ..."
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Cited by 9 (1 self)
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The parallel repetition theorem states that for any two provers one round game with value at most 1 − ɛ (for ɛ < 1/2), the value of the game repeated n times in parallel is at most (1−ɛ 3) Ω(n / log s) where s is the size of the answers set [Raz98],[Hol07]. For Projection Games the bound on the value of the game repeated n times in parallel was improved to (1−ɛ2) Ω(n) [Rao08] and was shown to be tight [Raz08]. In this paper we show that if the questions are taken according to a product distribution then the value of the repeated game is at most (1 − ɛ2 Ω(n / log s) and if in addition the game is a Projection Game we obtain a strong parallel repetition theorem, i.e., a bound of (1 − ɛ) Ω(n). 1