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Greedy strikes back: Improved facility location algorithms
 Journal of Algorithms
, 1999
"... A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the co ..."
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Cited by 183 (12 self)
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A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the commodities. We assume that the transportation costs form a metric. This problem is commonly referred to as the uncapacitated facility location (UFL) problem. Applications to bank account location and clustering, as well as many related pieces of work, are discussed by Cornuejols, Nemhauser and Wolsey [2]. Recently, the first constant factor approximation algorithm for this problem was obtained by Shmoys, Tardos and Aardal [16]. We show that a simple greedy heuristic combined with the algorithm by Shmoys, Tardos and Aardal, can be used to obtain an approximation guarantee of 2.408. We discuss a few variants of the problem, demonstrating better approximation factors for restricted versions of the problem. We also show that the problem is Max SNPhard. However, the inapproximability constants derived from the Max SNP hardness are very close to one. By relating this problem to Set Cover, we prove a lower bound of 1.463 on the best possible approximation ratio assuming NP / ∈ DT IME[n O(log log n)]. 1
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 116 (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.
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 101 (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.
Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs
, 1999
"... We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performa ..."
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Cited by 92 (6 self)
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We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performance ratio of 2  (1  o(1)) 2 ln ln \Delta ln \Delta for large \Delta, which improves the previously known ratio of 2 \Gamma log \Delta+O(1) \Delta obtained by Halldórsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For kuniform hypergraphs with n vertices, we achieve a ratio of k \Gamma (1 \Gamma o(1)) k ln ln n ln n for large n, and for kuniform hypergraphs with maximum degree at most \Delta, the algorithm achieves a ratio of k \Gamma (1 \Gamma o(1)) k(k\Gamma1) ln ln \Delta ln \Delta for large \Delta. These results considerably improve the previous best ratio of k(1\Gammac=\Delta 1 k\Gamma1 ) for bounded degree kuniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general kuniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldórsson.
A nearly lineartime approximation scheme for the Euclidean kmedian problem
, 1999
"... In the kmedian problem we are given a set N of n points in a metric space and a positive integer k: The objective is to locate k medians among the points so that the sum of the distances from each point in N to its closest median is minimized. The kmedian problem is a wellstudied, NPhard, bas ..."
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Cited by 38 (0 self)
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In the kmedian problem we are given a set N of n points in a metric space and a positive integer k: The objective is to locate k medians among the points so that the sum of the distances from each point in N to its closest median is minimized. The kmedian problem is a wellstudied, NPhard, basic clustering problem, which is closely related to facility location. Obtaining constantfactor approximations for this problem, even for the 2dimensional Euclidean metric, had long been an elusive goal. First Arora, Raghavan and Rao gave a randomized polynomialtime approximation scheme by extending techniques introduced originally by Arora for the Euclidean TSP. For any xed " > 0; their algorithm outputs a (1 + ")approximation in O(nkn log n) time.
An Improved Approximation Algorithm for the Metric Uncapacitated Facility Location Problem
, 2002
"... We design new approximation algorithm for the metric uncapacitated facility location problem. This algorithm is of LP rounding type and is based on a rounding technique developed in [57]. ..."
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Cited by 27 (3 self)
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We design new approximation algorithm for the metric uncapacitated facility location problem. This algorithm is of LP rounding type and is based on a rounding technique developed in [57].
KMedians, Facility Location, and the ChernoffWald Bound
, 2000
"... We study the general (nonmetric) facilitylocation and weighted kmedians problems, as well as the fractional facilitylocation and unweighted kmedians problems. We describe a natural randomized rounding scheme and use it to derive approximation algorithms for all of these problems. For facility l ..."
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Cited by 6 (1 self)
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We study the general (nonmetric) facilitylocation and weighted kmedians problems, as well as the fractional facilitylocation and unweighted kmedians problems. We describe a natural randomized rounding scheme and use it to derive approximation algorithms for all of these problems. For facility location and weighted kmedians, the respective algorithms are polynomialtime [Hk + d] and [(1 + )d; ln(n + n=)k]approximation algorithms. These performance guarantees improve on the best previous performance guarantees, due respectively to Hochbaum (1982) and Lin and Vitter (1992). For fractional kmedians, the algorithm is a new, Lagrangianrelaxation, [(1 + )d; (1 + )k]