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103
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 238 (32 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
Lectures on random planar curves and SchrammLoewner evolutions
, 2003
"... The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilist ..."
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Cited by 146 (8 self)
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The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilists and I therefore assume knowledge in stochastic calculus (Itô’s formula etc.), but no special background in basic complex analysis. These lecture notes are neither a book nor a compilation of research papers. While preparing them, I realized that it was hopeless to present all the recent results on this subject, or even to give the complete detailed proofs of a selected portion of them. Maybe this will disappoint part of the audience but the main goal of these lectures will be to try to transmit some ideas and heuristics. As a reader/part of an audience, I often think that omitting details is dangerous, and that ideas are sometimes better understood when the complete proofs are given, but in the present case, partly because the technicalities often use complex analysis tools that the audience might not be so
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 90 (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
Learning intersections and thresholds of halfspaces
, 2004
"... We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution on the Boolean hypercube to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog n ..."
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Cited by 88 (32 self)
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We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution on the Boolean hypercube to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog number of polynomialweight halfspaces under any distribution on the Boolean hypercube. As special cases of these results we obtain algorithms for learning intersections and thresholds of halfspaces. Our uniform distribution learning algorithms involve a novel nongeometric approach to learning halfspaces; we use Fourier techniques together with a careful analysis of the noise sensitivity of functions of halfspaces. Our algorithms for learning under any distribution use techniques from real approximation theory to construct lowdegree polynomial threshold functions. Finally, we also observe that any function of a constant number of polynomialweight halfspaces can be learned in polynomial time in the model of exact learning from membership and equivalence queries.
First Passage Percolation Has Sublinear Distance Variance
 Ann. Probab
, 1970
"... Let 0 < a < b < ∞, and for each edge e of Z d let ωe = a or ωe = b, each with probability 1/2, independently. This induces a random metric distω on the vertices of Z d, called first passage percolation. We prove that for d> 1 the distance distω(0,v) from the origin to a vertex v, v > ..."
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Cited by 55 (7 self)
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Let 0 < a < b < ∞, and for each edge e of Z d let ωe = a or ωe = b, each with probability 1/2, independently. This induces a random metric distω on the vertices of Z d, called first passage percolation. We prove that for d> 1 the distance distω(0,v) from the origin to a vertex v, v > 2, has variance bounded by C v/log v, where C = C(a,b,d) is a constant which may only depend on a, b and d. Some related variants are also discussed. 1
Conformally invariant scaling limits: an overview and collection of open problems
 Proceedings of the International Congress of Mathematicians, Madrid (M. SanzSolé et
, 2007
"... Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions beco ..."
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Cited by 52 (2 self)
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Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a oneparameter family of random curves called Stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
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 ..."
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Cited by 46 (13 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].
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 33 (10 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.