Results 1  10
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68
Maximizing nonmonotone submodular functions
 In Proceedings of 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2007
"... Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular fu ..."
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Cited by 85 (13 self)
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Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NPhard. In this paper, we design the first constantfactor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 2approximation and a randomizedapproximation algo
Vector Gaussian multiple description with individual and central receivers
 IEEE Trans. Information Theory
, 2007
"... The problem of L multiple descriptions of a stationary and ergodic Gaussian source with two levels of receivers is investigated. Each of the first level receivers receive (an arbitrary subset) k of the L descriptions, (k < L). The second level receiver receives all L descriptions. All the receivers, ..."
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Cited by 23 (3 self)
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The problem of L multiple descriptions of a stationary and ergodic Gaussian source with two levels of receivers is investigated. Each of the first level receivers receive (an arbitrary subset) k of the L descriptions, (k < L). The second level receiver receives all L descriptions. All the receivers, both at the first level and the second level, reconstruct the source using the subset of descriptions they receive. The corresponding reconstructions are subject to quadratic distortion constraints. Our main result is the derivation of an outer bound on the sum rate of the descriptions so that the distortion constraints are met. We show that a natural analogdigital separation architecture involving joint Gaussian vector quantizers and a binning scheme meets this outer bound with equality for several scenarios. These scenarios include the case when the distortion constraints are symmetric and the case for general distortion constraints with k = 2 and L = 3. We also show the robustness of this architecture: the distortions achieved are no larger when used to describe any nonGaussian source with the same covariance matrix. 1
Distances in random graphs with finite variance degrees
, 2008
"... In this paper we study a random graph with N nodes, where node j has degree Dj and {Dj} N j=1 are i.i.d. with P(Dj ≤ x) = F(x). We assume that 1 − F(x) ≤ cx −τ+1 for some τ> 3 and some constant c> 0. This graph model is a variant of the socalled configuration model, and includes heavy tail degree ..."
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Cited by 21 (11 self)
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In this paper we study a random graph with N nodes, where node j has degree Dj and {Dj} N j=1 are i.i.d. with P(Dj ≤ x) = F(x). We assume that 1 − F(x) ≤ cx −τ+1 for some τ> 3 and some constant c> 0. This graph model is a variant of the socalled configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like log ν N, when the base of the logarithm equals ν = E[Dj(Dj − 1)]/E[Dj]> 1. This confirms the heuristic argument of Newman, Strogatz and Watts [35]. In addition, the random fluctuations around this asymptotic mean log ν N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.
Properly colored subgraphs and rainbow subgraphs in edgecolorings with local constraints
 ALGORITHMS
, 2003
"... We consider a canonical Ramsey type problem. An edgecoloring of a graph is called mgood if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every mgood edgecoloring of K n yields a properly edgecolored ..."
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Cited by 21 (0 self)
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We consider a canonical Ramsey type problem. An edgecoloring of a graph is called mgood if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every mgood edgecoloring of K n yields a properly edgecolored copy of G, and let g(m, G) denote the smallest n such that every mgood edgecoloring of K n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G � K t, we have c 1mt 2 /ln t � f(m, K t) � c 2mt 2, and c � 1mt 3 /ln t � g(m, K t) � c � 2mt 3 /ln t, where c 1, c 2, c � 1, c � 2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) � n for all graphs G with n vertices and
Critical percolation on random regular graphs
, 2007
"... We describe the component sizes in critical independent pbond percolation on a random dregular graph on n vertices, where d ≥ 3 is fixed and n grows. We prove meanfield behavior around the critical probability pc = 1 d−1. In particular, we show that there is a scaling window of width n −1/3 aroun ..."
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Cited by 14 (6 self)
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We describe the component sizes in critical independent pbond percolation on a random dregular graph on n vertices, where d ≥ 3 is fixed and n grows. We prove meanfield behavior around the critical probability pc = 1 d−1. In particular, we show that there is a scaling window of width n −1/3 around pc in which the sizes of the largest components are roughly n 2/3 and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1 − ε(n))pc where ε(n) = o(1) but ε(n)n 1/3 → ∞, the sizes of the largest components are concentrated around an explicit function of n and ε(n) which is of order o(n 2/3). In the supercritical regime, i.e. p = (1 + ε(n))pc where ε(n) = o(1) but ε(n)n1/3 → ∞, the size of the largest component is concentrated around the value 2d d−2ε(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime.
Critical random graphs: diameter and mixing time
"... Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Worm ..."
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Cited by 13 (4 self)
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Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald [5]. These results extend to clusters of size n 2/3 of pbond percolation on any dregular nvertex graph where such clusters exist, provided that p(d − 1) ≤ 1 + O(n −1/3). 1.
Asymptotically the List Colouring Constants Are 1
 Journal of Combinatorial Theory Series B
"... In this paper we prove the following result about vertex list colourings, which shows that a conjecture from [9] is asymptotically correct. Let G be a graph with the sets of lists S(v), satisfying that for every vertex jS(v)j = (1+o(1))d and for each colour c 2 S(v), the number of neighbours of v th ..."
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Cited by 12 (4 self)
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In this paper we prove the following result about vertex list colourings, which shows that a conjecture from [9] is asymptotically correct. Let G be a graph with the sets of lists S(v), satisfying that for every vertex jS(v)j = (1+o(1))d and for each colour c 2 S(v), the number of neighbours of v that have c in their list is at most d. Then there exist a proper colouring of G from these lists.
Submodular function maximization via the multilinear relaxation and contention resolution schemes
 In ACM Symposium on Theory of Computing
, 2011
"... We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that all ..."
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Cited by 12 (1 self)
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We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a nonmonotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6, 22, 24, 13, 3], it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize