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117
A new greedy approach for facility location problems
"... We present a simple and natural greedy algorithm for the metric uncapacitated facility location problem achieving an approximation guarantee of 1.61 whereas the best previously known was 1.73. Furthermore, we will show that our algorithm has a property which allows us to apply the technique of Lagra ..."
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Cited by 132 (9 self)
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We present a simple and natural greedy algorithm for the metric uncapacitated facility location problem achieving an approximation guarantee of 1.61 whereas the best previously known was 1.73. Furthermore, we will show that our algorithm has a property which allows us to apply the technique of Lagrangian relaxation. Using this property, we can nd better approximation algorithms for many variants of the facility location problem, such as the capacitated facility location problem with soft capacities and a common generalization of the kmedian and facility location problem. We will also prove a lower bound on the approximability of the kmedian problem.
Adwords and generalized online matching
 In FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competit ..."
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Cited by 130 (6 self)
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How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1 − 1/e for this problem. 1
Greedy Facility Location Algorithms analyzed using Dual Fitting with FactorRevealing LP
 Journal of the ACM
, 2001
"... We present a natural greedy algorithm for the metric uncapacitated facility location problem and use the method of dual fitting to analyze its approximation ratio, which turns out to be 1.861. The running time of our algorithm is O(m log m), where m is the total number of edges in the underlying c ..."
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Cited by 125 (13 self)
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We present a natural greedy algorithm for the metric uncapacitated facility location problem and use the method of dual fitting to analyze its approximation ratio, which turns out to be 1.861. The running time of our algorithm is O(m log m), where m is the total number of edges in the underlying complete bipartite graph between cities and facilities. We use our algorithm to improve recent results for some variants of the problem, such as the fault tolerant and outlier versions. In addition, we introduce a new variant which can be seen as a special case of the concave cost version of this problem.
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 101 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
New upper bounds on sphere packings
, 2001
"... Abstract. We develop an analogue for sphere packing of the linear programming bounds for errorcorrecting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to s ..."
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Cited by 52 (6 self)
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Abstract. We develop an analogue for sphere packing of the linear programming bounds for errorcorrecting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24.
Universally optimal distribution of points on spheres
 Journal of the American Mathematical Society
"... Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a ..."
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Cited by 44 (8 self)
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Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m −1)design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E8 and Leech lattices. We also prove the same result for the vertices of the 600cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact twopoint homogeneous spaces, and we
Upper Bounds for ConstantWeight Codes
 IEEE TRANS. INFORM. THEORY
, 2000
"... Let A(n; d; w) denote the maximum possible number of codewords in an (n; d; w) constantweight binary code. We improve upon the best known upper bounds on A(n; d; w) in numerous instances for n 6 24 and d 6 12, which is the parameter range of existing tables. Most improvements occur for d = 8; 10, ..."
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Cited by 44 (1 self)
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Let A(n; d; w) denote the maximum possible number of codewords in an (n; d; w) constantweight binary code. We improve upon the best known upper bounds on A(n; d; w) in numerous instances for n 6 24 and d 6 12, which is the parameter range of existing tables. Most improvements occur for d = 8; 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n 6 28 and d 6 14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n; d; w) by means of mapping constantweight codes into Euclidean space. This approach produces, among other results, a bound on A(n; d; w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doublyconstantweight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doublyboundedweight codes, which may be thought of as a generaliz...
Constrained Energy Problems with Applications to Orthogonal Polynomials of a Discrete Variable
 J. Anal. Math
, 1997
"... Given a positive measure oe with koek ? 1 we write ¯ 2 M oe if ¯ is a probability measure and oe \Gamma ¯ is a positive measure. Under some general assumptions on the constraining measure oe and a weight function w we prove existence and uniqueness of a measure oe w that minimizes the weighted ..."
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Cited by 41 (6 self)
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Given a positive measure oe with koek ? 1 we write ¯ 2 M oe if ¯ is a probability measure and oe \Gamma ¯ is a positive measure. Under some general assumptions on the constraining measure oe and a weight function w we prove existence and uniqueness of a measure oe w that minimizes the weighted logarithmic energy over the class M oe . We also obtain a characterization theorem, a saturation result and a balayage representation for the measure oe w . As applications of our results we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. 1 Introduction In this paper we shall investigate constrained energy problems in the presence of an external field. Before defining the problem, we briefly recall the classical and the weighted energy problems of potential theory. In so doing, we introduce the terminology that will be needed for statin...
Covering arrays and intersecting codes
 Journal of Combinatorial Designs
, 1993
"... A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rkn ..."
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Cited by 38 (0 self)
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A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rknyi, Katona, and Kleitman and Spencer. The present article is concerned with the case t = 3, where important (but unpublished) contributions were made by Busschbach and Roux in the 1980s. One of the principal constructions makes use of intersecting codes (linear codes with the property that any two nonzero codewords meet). This article studies the properties of 3covering arrays and intersecting codes, and gives a table of the best 3covering arrays presently known. For large n the minimal k satisfies 3.21256 < k / log n < 7.56444. 01993