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47
Optimal algorithms and inapproximability results for every CSP
 In Proc. 40 th ACM STOC
, 2008
"... Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the ..."
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Cited by 144 (13 self)
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Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result: If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomialtime algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut
On the Hardness of Approximating Multicut and SparsestCut
 In Proceedings of the 20th Annual IEEE Conference on Computational Complexity
, 2005
"... We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1. ..."
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Cited by 102 (5 self)
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We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.
ANGULAR SYNCHRONIZATION BY EIGENVECTORS AND SEMIDEFINITE PROGRAMMING: ANALYSIS AND APPLICATION TO CLASS AVERAGING IN CRYOELECTRON MICROSCOPY
, 905
"... Abstract. The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ1,..., θn from m noisy measurements of their offsets θi − θj mod 2π. Of particular interest is angle recovery in the presence of many outlier measurements ..."
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Cited by 43 (17 self)
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Abstract. The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ1,..., θn from m noisy measurements of their offsets θi − θj mod 2π. Of particular interest is angle recovery in the presence of many outlier measurements that are uniformly distributed in [0,2π) and carry no information on the true offsets. We introduce an efficient recovery algorithm for the unknown angles from the top eigenvector of a specially designed Hermitian matrix. The eigenvector method is extremely stable and succeeds even when the number of outliers is exceedingly large. For example, we successfully estimate n = 400 angles from a full set of m = `400 ´ offset measurements of which 90 % are outliers in less than a second 2 on a commercial laptop. We use random matrix theory to prove that the eigenvector method q gives
Parallel repetition in projection games and a concentration bound
 In Proc. 40th STOC
, 2008
"... In a two player game, a referee asks two cooperating players (who are not allowed to communicate) questions sampled from some distribution and decides whether they win or not based on some predicate of the questions and their answers. The parallel repetition of the game is the game in which the refe ..."
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Cited by 42 (8 self)
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In a two player game, a referee asks two cooperating players (who are not allowed to communicate) questions sampled from some distribution and decides whether they win or not based on some predicate of the questions and their answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than (1 − ǫ), what’s the best they can do in the repeated game? We improve earlier results [Raz98, Hol07], which showed that the players cannot win all copies in the repeated game with probability better than (1 −ǫ 3) Ω(n/c) (here c is the length of the answers in the game), in the following ways: • We prove the bound (1 −ǫ 2) Ω(n) as long as the game is a “projection game”, the type of game most commonly used in hardness of approximation results. Our bound is independent of the answer length and has a better dependence on ǫ. By the recent work of Raz [Raz08], this bound is tight. A consequence of this bound is that the Unique Games Conjecture of Khot [Kho02] is equivalent to: Unique Games Conjecture There is an unbounded increasing function f: R + → R + such that for every ǫ> 0, there exists an alphabet size M(ǫ) for which it is NPhard to distinguish a Unique Game with alphabet size M in which a 1 −ǫ 2 fraction of the constraints can be satisfied from one in which a 1 − ǫf(1/ǫ) fraction of the constraints can be satisfied. • We prove a concentration bound for parallel repetition (of general games) showing that for any constant 0 < δ < ǫ, the probability that the players win a (1 −ǫ+δ) fraction of the games in the parallel repetition is at most exp � −Ω(δ 4 n/c) �. An application of this is in testing Bell Inequalities. Our result implies that the parallel repetition of the CHSH game can be used to get an experiment that has a very large classical versus quantum gap.
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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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 29 (8 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
How to Round Any CSP
"... A large number of interesting combinatorial optimization ..."
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Cited by 26 (3 self)
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A large number of interesting combinatorial optimization
Improved approximation guarantees through higher levels of SDP hierarchies
 In Proceedings of the 11th International Workshop, APPROX
, 2008
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How to play unique games using embeddings
 In Proc. Annual IEEE Symposium on Foundations of Computer Science
, 2006
"... In this paper we present a new approximation algorithm for Unique Games. For a Unique Game with n vertices and k states (labels), if a (1 − ε) fraction of all constraints is satisfiable, the algorithm finds an assignment satisfying a 1 − O(ε √ log n log k) fraction of all constraints. To this end, w ..."
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Cited by 20 (3 self)
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In this paper we present a new approximation algorithm for Unique Games. For a Unique Game with n vertices and k states (labels), if a (1 − ε) fraction of all constraints is satisfiable, the algorithm finds an assignment satisfying a 1 − O(ε √ log n log k) fraction of all constraints. To this end, we introduce new embedding techniques for rounding semidefinite relaxations of problems with large domain size. 1
Rounding Parallel Repetitions of Unique Games
"... We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically, denoting by val(G) the value of a twoprover unique game G, and by sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every ..."
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Cited by 15 (5 self)
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We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically, denoting by val(G) the value of a twoprover unique game G, and by sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every ℓ ∈ N, if sdpval(G) � 1 − δ, then val(G ℓ) � 1 − √ sℓδ. Here, G ℓ denotes the ℓfold parallel repetition of G, and s = O(log ( k/δ)), where k denotes the alphabet size of the game. For the special case where G is an XOR game (i.e., k = 2), we obtain the same bound but with s as an absolute constant. Our bounds on s are optimal up to a factor of O(log ( 1/δ)). For games with a significant gap between the quantities val(G) and sdpval(G), our result implies that val(G ℓ) may be much larger than val(G) ℓ, giving a counterexample to the strong parallel repetition conjecture. In a recent breakthrough, Raz (FOCS ’08) has shown such an example using the maxcut game on odd cycles. Our results are based on a generalization of his techniques.