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51
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 86 (12 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
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
"... In this paper, we present improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*ast ..."
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Cited by 59 (8 self)
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In this paper, we present improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*astad's celebrated result [13] shows that the maximumclique size in a graph with n vertices is inapproximable inpolynomial time within a factor n1ffl for arbitrarily smallconstant ffl> 0 unless NP=ZPP. In this paper, we aimat getting the best subconstant value of ffl in H*astad's result. We prove that clique size is inapproximable within a factor n2(log n)1fl (corresponding to ffl = 1(log n)fl) for some constant fl> 0 unless NP ` ZPTIME(2(log n) O(1)). This improves the previous best inapproximability factor of
Gaussian Bounds for Noise Correlation of Functions and Tight Analysis of Long Codes
 In IEEE Symposium on Foundations of Computer Science (FOCS
, 2008
"... In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recentl ..."
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Cited by 35 (5 self)
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In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O’Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multidimensional Gaussian distributions. We present two applications of the new bounds to the theory of social choice. We show that Majority is asymptotically the most predictable function among all low influence functions given a random sample of the voters. Moreover, we derive an almost tight bound in the context of Condorcet aggregation and low influence voting schemes on a large number of candidates. In particular, we show that for every low influence aggregation function, the probability that Condorcet voting on k candidates will result in a unique candidate that is preferable to all others is k−1+o(1). This matches the asymptotic behavior of the majority function for which the probability is k−1−o(1). A number of applications in hardness of approximation in theoretical computer science were
Approximation algorithms for unique games
 In FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
"... Abstract: A unique game is a type of constraint satisfaction problem with two variables per constraint. The value of a unique game is the fraction of the constraints satisfied by an optimal solution. Khot (STOC’02) conjectured that for arbitrarily small γ,ε> 0 it is NPhard to distinguish games of ..."
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Cited by 33 (0 self)
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Abstract: A unique game is a type of constraint satisfaction problem with two variables per constraint. The value of a unique game is the fraction of the constraints satisfied by an optimal solution. Khot (STOC’02) conjectured that for arbitrarily small γ,ε> 0 it is NPhard to distinguish games of value smaller than γ from games of value larger than 1 − ε. Several recent inapproximability results rely on Khot’s conjecture. Considering the case of subconstant ε, Khot (STOC’02) analyzes an algorithm based on semidefinite programming that satisfies a constant fraction of the constraints in unique games of value 1 − O(k−10 · (logk) −5), where k is the size of the domain of the variables. We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1−O(1/logn), satisfies a constant fraction of the constraints, where n is the number of variables. This is an improvement over Khot’s algorithm if the domain is sufficiently large.
Towards Sharp Inapproximability For Any 2CSP
, 2008
"... 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 22 (2 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. Thisimproves 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 20 (4 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
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 17 (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
Unconditional pseudorandom generators for low degree polynomials
, 2007
"... We give an explicit construction of pseudorandom generators against low degree polynomials over finite fields. We show that the sum of 2d smallbiased generators with error ɛ2O(d) is a pseudorandom generator against degree d polynomials with error ɛ. This gives a generator with seed length 2O(d) log ..."
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Cited by 16 (3 self)
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We give an explicit construction of pseudorandom generators against low degree polynomials over finite fields. We show that the sum of 2d smallbiased generators with error ɛ2O(d) is a pseudorandom generator against degree d polynomials with error ɛ. This gives a generator with seed length 2O(d) log (n/ɛ). Our construction follows the recent breakthrough result of Bogadnov and Viola [BV07]. Their work shows that the sum of d smallbiased generators is a pseudorandom generator against degree d polynomials, assuming the Inverse Gowers Conjecture. However, this conjecture is only proven for d = 2, 3. The main advantage of our work is that it does not rely on any unproven conjectures. 1
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 15 (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
On the Approximation Resistance of a Random Predicate
"... A predicate is approximation resistant if no probabilistic polynomial time approximation algorithm can do significantly better then the naive algorithm that picks an assignment uniformly at random. Assuming that the Unique Games Conjecture is true we prove that most Boolean predicates are approxima ..."
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Cited by 13 (1 self)
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A predicate is approximation resistant if no probabilistic polynomial time approximation algorithm can do significantly better then the naive algorithm that picks an assignment uniformly at random. Assuming that the Unique Games Conjecture is true we prove that most Boolean predicates are approximation resistant.