Results 1  10
of
43
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 173 (24 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
Sharp Thresholds of Graph properties, and the ksat Problem
 J. Amer. Math. Soc
, 1998
"... Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a nonsharp thres ..."
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Cited by 165 (7 self)
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Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a nonsharp threshold), then there is a list of graphs of bounded size such that P can be approximated by the property of having one of the graphs as a subgraph. One striking consequences of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of n.
Vertex Cover Might be Hard to Approximate to within 2ɛ
 IN PROCEEDINGS OF THE 18TH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2003
"... We show that vertex cover is hard to approximate within any constant factor better than 2 where the hardness is based on a conjecture regarding the power of unique 2prover1round games presented in [15]. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k ..."
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Cited by 119 (11 self)
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We show that vertex cover is hard to approximate within any constant factor better than 2 where the hardness is based on a conjecture regarding the power of unique 2prover1round games presented in [15]. 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.
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
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 73 (4 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.
Noise sensitivity of Boolean functions and applications to percolation, Inst. Hautes Études
, 1999
"... It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority ..."
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Cited by 72 (16 self)
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It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noisestable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an n + 1 by n grid. A configuration is a function that assigns to every edge the value 0 or 1. Let ω be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges e with ω(e) = 1. By duality, the probability for having a crossing is 1/2. Fix an ǫ ∈ (0,1). For each edge e, let ω ′ (e) = ω(e) with probability 1 − ǫ, and ω ′ (e) = 1 − ω(e)
IMPROVED LOWER BOUNDS FOR EMBEDDINGS INTO L1
 SIAM J. COMPUT.
, 2009
"... We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bo ..."
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Cited by 32 (5 self)
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We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a wellknown semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of (log log n) 1/6−o(1) due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l1, in Proceedings of the 46th Annual IEEE Symposium
Some Topics in Analysis of Boolean Functions
"... This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an exten ..."
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Cited by 29 (0 self)
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This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an extension of Chernoff/Hoeffding bounds to higherdegree polynomials; and, hardness for approximation algorithms.
Graph Products, Fourier Analysis and Spectral Techniques
, 2003
"... We consider powers of regular graphs defined by the weak graph product and give a characterization of maximumsize independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In man ..."
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Cited by 24 (7 self)
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We consider powers of regular graphs defined by the weak graph product and give a characterization of maximumsize independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the independent sets induced by the base graph are the only maximumsize independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size. Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on f0; : : : ; r \Gamma 1gn, generalizing some useful results from the f0; 1gn case.