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Vertex Cover Might be Hard to Approximate to within 2  ɛ
"... Based on a conjecture regarding the power of unique 2prover1round games presented in [Khot02], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on kuniform hypergraph ..."
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Cited by 151 (11 self)
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Based on a conjecture regarding the power of unique 2prover1round games presented in [Khot02], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on kuniform hypergraphs is hard to approximate within any constant factor better than k.
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 55 (6 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
Nearoptimal algorithms for Unique Games
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Unique games are constraint satisfaction problems that can be viewed as a generalization of MaxCut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none ar ..."
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Cited by 46 (8 self)
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Unique games are constraint satisfaction problems that can be viewed as a generalization of MaxCut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none are satisfiable. It has been shown to imply a number of inapproximability results for fundamental problems that seem difficult to obtain by more standard complexity assumptions. Thus, proving or refuting this conjecture is an important goal. We present significantly improved approximation algorithms for unique games. For instances with domain size k where the optimal solution satisfies 1 − ε fraction of all constraints, our algorithms satisfy roughly k −ε/(2−ε) and 1 − O ( √ ε log k) fraction of all constraints. Our algorithms are based on rounding a natural semidefinite programming relaxation for the problem and their performance almost matches the integrality gap of this relaxation. Our results are near optimal if the Unique Games Conjecture is true, i.e. any improvement (beyond low order terms) would refute the conjecture. 1
A brief introduction to Fourier analysis on the Boolean cube
 Theory of Computing Library– Graduate Surveys
, 2008
"... Abstract: We give a brief introduction to the basic notions of Fourier analysis on the ..."
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Cited by 34 (4 self)
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Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
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 33 (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
On the hardness of approximating the network coding capacity
 In Proc. IEEE Symp. on Inform. Theory (ISIT
, 2008
"... Abstract—This work addresses the computational complexity of achieving the capacity of a general network coding instance. We focus on the linear capacity, namely the capacity of the given instance when restricted to linear encoding functions. It has been shown [Lehman and Lehman, SODA 2005] that de ..."
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Cited by 32 (8 self)
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Abstract—This work addresses the computational complexity of achieving the capacity of a general network coding instance. We focus on the linear capacity, namely the capacity of the given instance when restricted to linear encoding functions. It has been shown [Lehman and Lehman, SODA 2005] that determining the (scalar) linear capacity of a general network coding instance is NPhard. In this work we initiate the study of approximation in this context. Namely, we show that given an instance to the general network coding problem of linear capacity C, constructing a linear code of rate αC for any universal (i.e., independent of the size of the instance) constant α ≤ 1 is “hard”. Specifically, finding such network codes would solve a long standing open problem in the field of graph coloring. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., a general instance in which the underlying graph is planar). We show that even for planar networks this problem remains NPhard. I.
New approximation guarantee for chromatic number
 STOC'06
, 2006
"... We describe how to color every 3colorable graph with O(n0.2111) colors, thus improving an algorithm of Blum and Karger from almost a decade ago. Our analysis uses new geometric ideas inspired by the recent work of Arora, Rao, and Vazirani on SPARSEST CUT, and these ideas show promise of leading to ..."
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Cited by 31 (3 self)
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We describe how to color every 3colorable graph with O(n0.2111) colors, thus improving an algorithm of Blum and Karger from almost a decade ago. Our analysis uses new geometric ideas inspired by the recent work of Arora, Rao, and Vazirani on SPARSEST CUT, and these ideas show promise of leading to further improvements.
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 29 (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.
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
Nonnegative Weighted #CSPs: An Effective Complexity Dichotomy
"... We prove a complexity dichotomy theorem for all nonnegative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms [19, 8, 18, 12] and the celebrated dichotomy theorem for unwe ..."
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Cited by 16 (9 self)
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We prove a complexity dichotomy theorem for all nonnegative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms [19, 8, 18, 12] and the celebrated dichotomy theorem for unweighted #CSP [6, 4, 21, 22]. Our dichotomy theorem gives a succinct criterion for tractability. If a set F of constraint functions satisfies the criterion, then the counting CSP problem defined by F is solvable in polynomial time; if it does not satisfy the criterion, then the problem is #Phard. We furthermore show that the question of whether F satisfies the criterion is decidable in NP. Surprisingly, our tractability criterion is simpler than the previous criteria for the more restricted classes of problems, although when specialized to those cases, they are logically equivalent. Our proof mainly uses Linear