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30
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
Noise stability of functions with low influences: invariance and optimality
"... In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions f: {−1, 1} n → {−1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of proba ..."
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Cited by 127 (17 self)
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In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions f: {−1, 1} n → {−1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known nonlinear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly “smoothed”; this extension is essential for our applications to “noise stability”type problems. In particular, as applications of the invariance principle we prove two conjectures: the “Majority Is Stablest ” conjecture [27] from theoretical computer science, which was the original motivation for this work, and the “It Ain’t Over Till It’s Over” conjecture [25] from social choice theory. The “Majority Is Stablest ” conjecture and its generalizations proven here in conjunction with “Unique Games” and its variants imply a number of (optimal) inapproximability results for graph problems.
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.
Towards Sharp Inapproximability For Any 2CSP
"... We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good ..."
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Cited by 32 (1 self)
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We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clausebyclause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are “hard ” to round in a certain sense, we obtain a Unique Gamesbased inapproximability matching this “hardness ” of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the “hardness ” of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2CSPs. As an application, we show that MAX 2AND is hard to approximate within 0.87435. This improves upon the best previous hardness of αGW + ɛ ≈ 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of MAX 2AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor αGW.
Approximation Resistant Predicates From Pairwise Independence
, 2008
"... We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwis ..."
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Cited by 32 (5 self)
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We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q] k whose support is contained in the set of satisfying assignments to P. Using constructions of pairwise independent distributions this result implies that • For general k ≥ 3 and q ≥ 2, theMAX kCSPq problem is UGhard to approximate within O(kq 2)/q k + ɛ. • For the special case of q =2, i.e., boolean variables, we can sharpen this bound to (k + O(k 0.525))/2 k + ɛ, improving upon the best previous bound of 2k/2 k +ɛ (Samorodnitsky and Trevisan, STOC’06) by essentially a factor 2. • Finally, again for q =2, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the O(k 0.525) term can be replaced by the constant 4. 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
Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs
"... We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Gameshard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set ..."
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Cited by 21 (0 self)
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We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Gameshard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set is Unique Gameshard to approxid mate to within a factor O( log2). This improves the d d logO(1) Unique Games hardness result of Samorod
Nearoptimal algorithms for maximum constraint satisfaction problems
 In SODA ’07: Proceedings of the eighteenth annual ACMSIAM symposium on Discrete algorithms
, 2007
"... In this paper we present approximation algorithms for the maximum constraint satisfaction problem with k variables in each constraint (MAX kCSP). Given a (1 − ε) satisfiable 2CSP our first algorithm finds an assignment of variables satisfying a 1 − O ( √ ε) fraction of all constraints. The best pr ..."
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Cited by 20 (3 self)
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In this paper we present approximation algorithms for the maximum constraint satisfaction problem with k variables in each constraint (MAX kCSP). Given a (1 − ε) satisfiable 2CSP our first algorithm finds an assignment of variables satisfying a 1 − O ( √ ε) fraction of all constraints. The best previously known result, due to Zwick, was 1 − O(ε 1/3). The second algorithm finds a ck/2 k approximation for the MAX kCSP problem (where c> 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which had an approximation guarantee of Ω(k/(2 k log k)). Both results are optimal assuming the Unique Games Conjecture and are based on rounding natural semidefinite programming relaxations. We also believe that our algorithms and their analysis are simpler than those previously known. 1
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 15 (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
Maximally stable Gaussian partitions with discrete applications
 Israel J. Math
"... Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat k ..."
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Cited by 13 (3 self)
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Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat kernel and derive as applications: • An optimality result for majority in the context of Condorcet voting. • A proof of a conjecture on “cosmic coin tossing ” for low influence functions. We also discuss a Gaussian noise stability conjecture which may be viewed as a generalization of the “Double Bubble ” theorem and show that it implies: • A proof of the “Plurality is Stablest Conjecture”. • That the FriezeJerrum SDP for MAXqCUT achieves the optimal approximation factor assuming the Unique Games Conjecture.