Results 1  10
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46
The Importance of Being Biased
, 2002
"... The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1 ..."
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Cited by 86 (8 self)
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The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1
New results for learning noisy parities and halfspaces
 In Proceedings of the 47th Annual Symposium on Foundations of Computer Science (FOCS
, 2006
"... We address wellstudied problems concerning the learnability of parities and halfspaces in the presence of classification noise. Learning of parities under the uniform distribution with random classification noise, also called the noisy parity problem is a famous open problem in computational learni ..."
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Cited by 47 (11 self)
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We address wellstudied problems concerning the learnability of parities and halfspaces in the presence of classification noise. Learning of parities under the uniform distribution with random classification noise, also called the noisy parity problem is a famous open problem in computational learning. We reduce a number of basic problems regarding learning under the uniform distribution to learning of noisy parities. We show that under the uniform distribution, learning parities with adversarial classification noise reduces to learning parities with random classification noise. Together with the parity learning algorithm of Blum et al. [5], this gives the first nontrivial algorithm for learning parities with adversarial noise. We show that learning of DNF expressions reduces to learning noisy parities of just logarithmic number of variables. We show that learning of kjuntas reduces to learning noisy parities of k variables. These reductions work even in the presence of random classification noise in the original DNF or junta. We then consider the problem of learning halfspaces over Qn with adversarial noise or finding a halfspace that maximizes the agreement rate with a given set of examples. We prove an essentially optimal hardness factor of 2 − ɛ, improving the factor of 85 84 − ɛ due to Bshouty and Burroughs [8]. Finally, we show that majorities of halfspaces are hard to PAClearn using any representation, based on the cryptographic assumption underlying the AjtaiDwork cryptosystem.
Conditional hardness for approximate coloring
 In STOC 2006
, 2006
"... We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = ..."
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Cited by 38 (12 self)
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We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST3COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noisestability quantities using the invariance principle of Mossel et al [MOO’05].
Ruling out PTAS for graph minbisection, dense ksubgraph, and bipartite clique
 SIAM J. Comput
"... Abstract Assuming that NP 6 ` "ffl?0 BPTIME(2nffl), we show that Graph MinBisection, Dense kSubgraph and Bipartite Clique have no Polynomial Time Approximation Scheme (PTAS). We give a reduction from the Minimum Distance of Code Problem (MDC). Starting with an instance of MDC, we build a Quasi ..."
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Cited by 30 (0 self)
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Abstract Assuming that NP 6 ` "ffl?0 BPTIME(2nffl), we show that Graph MinBisection, Dense kSubgraph and Bipartite Clique have no Polynomial Time Approximation Scheme (PTAS). We give a reduction from the Minimum Distance of Code Problem (MDC). Starting with an instance of MDC, we build a Quasirandom PCP that suffices to prove the desired inapproximability results. In a Quasirandom PCP, the query pattern of the verifier looks random in certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial and it can be checked probabilistically by reading a constant number of its values.
Hardness Results for Coloring 3Colorable 3Uniform Hypergraphs
, 2002
"... In this paper, we consider the problem of coloring a 3colorable 3uniform hypergraph. In the minimization version of this problem, given a 3colorable 3uniform hypergraph, one seeks an algorithm tocolor the hypergraph with as few colors as possible. We show that it is NPhard to color a 3colorab ..."
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Cited by 20 (9 self)
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In this paper, we consider the problem of coloring a 3colorable 3uniform hypergraph. In the minimization version of this problem, given a 3colorable 3uniform hypergraph, one seeks an algorithm tocolor the hypergraph with as few colors as possible. We show that it is NPhard to color a 3colorable 3uniform hypergraph with constantly many colors. In fact, we show a stronger result that it is NPhard to distinguish whether a 3uniform hypergraph with n vertices is 3colorable or it contains no independentset of size ffin for an arbitrarily small constant ffi? 0. In the maximization version of the problem, givena 3uniform hypergraph, the goal is to color the vertices with 3 colors so as to maximize the number ofnonmonochromatic edges. We show that it is NPhard to distinguish whether a 3uniform hypergraphis 3colorable or any coloring of the vertices with 3 colors has at most 89 + ffl fraction of the edges nonmonochromatic where ffl? 0 is an arbitrarily small constant. This result is tight since assigning a randomcolor independently to every vertex makes 8 9 fraction of the edges nonmonochromatic.These results are obtained via a new construction of a probabilistically checkable proof system (PCP) for NP. We develop a new construction of the PCP Outer Verifier. An important feature of this construction is smoothening of the projection maps. We believe that the techniques in this paper would be quite useful in future. As an application of ourtechniques, we give a simpler proof of H*astad's result [11] that for every constant ffl? 0, it is NPhardto distinguish satisfiable instances of Max3SAT from instances where no assignment satisfies more that 78 + ffl fraction of the clauses. Dinur, Regev and Smyth [6] independently showed that it is NPhard to color a 2colorable 3uniformhypergraph with constantly many colors. In the "good case", the hypergraph they construct is 2colorableand hence their result is stronger. In the "bad case " however, the hypergraph we construct has a stronger property, namely, it does not even contain an independent set of size ffin.
A New Trust Region Technique for the Maximum Weight Clique Problem
 Discrete Applied Mathematics
, 2002
"... A new simple generalization of the MotzkinStraus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objecti ..."
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Cited by 19 (2 self)
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A new simple generalization of the MotzkinStraus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a MotzkinStraus optimum coincides with such a point. The developed method has complexity O(n ), where n is the number of graph vertices. It was implemented in a publicly available software package QUALEXMS.
Query efficient PCPs with perfect completeness
 In 42nd Annual Symposium on Foundations of Computer Science
, 2001
"... For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability ..."
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Cited by 16 (2 self)
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For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability at most 2 k 2 +1 . In particular, the verifier achieves optimal amortized query complexity of 1 + for arbitrarily small constant > 0. Such a characterization was already proved by Samorodnitsky and Trevisan [15], but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier we can decrease the number of query bits to 2k + k 2 , the same number obtained in [15]. Finally we extend some of the results to larger domains. Royal Institute of Technology, Stockholm, work done while visiting Institute for Advanced Study, supported by NSF grant CCR9987077.
Graphs with tiny vector chromatic numbers and huge chromatic numbers
 SIAM J. Comput
"... Abstract. Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246–265] introduced the notion of a vector coloring of a graph. In particular, they showed that every kcolorable graph is also vector kcolorable, and that for constant k, graphs that are vector kcolorable can be colored by roughly ∆ 1−2 ..."
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Cited by 14 (2 self)
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Abstract. Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246–265] introduced the notion of a vector coloring of a graph. In particular, they showed that every kcolorable graph is also vector kcolorable, and that for constant k, graphs that are vector kcolorable can be colored by roughly ∆ 1−2/k colors. Here ∆ is the maximum degree in the graph and is assumed to be of the order of n δ for some 0 <δ<1. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector kcolorable but do not have independent sets significantly larger than n/ ∆ 1−2/k (and hence cannot be colored with significantly fewer than ∆ 1−2/k colors). For k = O(log n / log log n) we show vector kcolorable graphs that do not have independent sets of size (log n) c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn. As part of our proof, we analyze “property testing ” algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are “far ” from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653–750] for this problem.
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 14 (0 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
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 14 (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