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Approximation algorithms for metric facility location and kmedian problems using the . . .
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Approximation Algorithms for Projective Clustering
 Proceedings of the ACM SIGMOD International Conference on Management of data, Philadelphia
, 2000
"... We consider the following two instances of the projective clustering problem: Given a set S of n points in R d and an integer k ? 0; cover S by k hyperstrips (resp. hypercylinders) so that the maximum width of a hyperstrip (resp., the maximum diameter of a hypercylinder) is minimized. Let w ..."
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Cited by 302 (22 self)
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We consider the following two instances of the projective clustering problem: Given a set S of n points in R d and an integer k ? 0; cover S by k hyperstrips (resp. hypercylinders) so that the maximum width of a hyperstrip (resp., the maximum diameter of a hypercylinder) is minimized. Let w be the smallest value so that S can be covered by k hyperstrips (resp. hypercylinders), each of width (resp. diameter) at most w : In the plane, the two problems are equivalent. It is NPHard to compute k planar strips of width even at most Cw ; for any constant C ? 0 [50]. This paper contains four main results related to projective clustering: (i) For d = 2, we present a randomized algorithm that computes O(k log k) strips of width at most 6w that cover S. Its expected running time is O(nk 2 log 4 n) if k 2 log k n; it also works for larger values of k, but then the expected running time is O(n 2=3 k 8=3 log 4 n). We also propose another algorithm that computes a c...
A constantfactor approximation algorithm for the kmedian problem
 In Proceedings of the 31st Annual ACM Symposium on Theory of Computing
, 1999
"... We present the first constantfactor approximation algorithm for the metric kmedian problem. The kmedian problem is one of the most wellstudied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are re ..."
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Cited by 249 (13 self)
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We present the first constantfactor approximation algorithm for the metric kmedian problem. The kmedian problem is one of the most wellstudied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are relatively close with respect to some measure. For the metric kmedian problem, we are given n points in a metric space. We select k of these to be cluster centers, and then assign each point to its closest selected center. If point j is assigned to a center i, the cost incurred is proportional to the distance between i and j. The goal is to select the k centers that minimize the sum of the assignment costs. We give a 6 2 3approximation algorithm for this problem. This improves upon the best previously known result of O(log k log log k), which was obtained by refining and derandomizing a randomized O(log n log log n)approximation algorithm of Bartal. 1
Improved Combinatorial Algorithms for the Facility Location and kMedian Problems
 In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science
, 1999
"... We present improved combinatorial approximation algorithms for the uncapacitated facility location and kmedian problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414 ..."
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Cited by 225 (12 self)
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We present improved combinatorial approximation algorithms for the uncapacitated facility location and kmedian problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414 + in ~ O(n 2 =) time. This also yields a bicriteria approximation tradeoff of (1 +; 1+ 2=) for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to Jain and Vazirani, we get an approximation ratio of 1.853 in ~ O(n 3 ) time. This is already very close to the approximation guarantee of the best known algorithm which is LPbased. Further, combined with the best known LPbased algorithm for facility location, we get a very slight improvement in the approximation factor for facility location, achieving 1.728....
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 210 (11 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
Influence Sets Based on Reverse Nearest Neighbor Queries
 In SIGMOD
, 2000
"... Inherent in the operation of many decision support and continuous referral systems is the notion of the "influence" of a data point on the database. This notion arises in examples such as finding the set of customers affected by the opening of a new store outlet location, notifying the sub ..."
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Cited by 148 (1 self)
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Inherent in the operation of many decision support and continuous referral systems is the notion of the "influence" of a data point on the database. This notion arises in examples such as finding the set of customers affected by the opening of a new store outlet location, notifying the subset of subscribers to a digital library who will find a newly added document most relevant, etc. Standard approaches to determining the influence set of a data point involve range searching and nearest neighbor queries. In this paper, we formalize a novel notion of influence based on reverse neighbor queries and its variants. Since the nearest neighbor relation is not symmetric, the set of points that are closest to a query point (i.e., the nearest neighbors) differs from the set of points that have the query point as their nearest neighbor (called the reverse nearest neighbors). Influence sets based on reverse nearest neighbor (RNN) queries seem to capture the intuitive notion of influence from our ...
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 146 (12 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.
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 145 (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.