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44
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 253 (12 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 227 (11 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....
An approximation algorithm for the generalized assignment problem
, 1993
"... The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing job j on machine i requires time pif and incurs a cost of c,f, each machine / is available for 7", time units, ..."
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Cited by 202 (5 self)
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The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing job j on machine i requires time pif and incurs a cost of c,f, each machine / is available for 7&quot;, time units, and the objective is.t»minimize the total cost incurred. Our main result is as follows. There is a polynomialtime algorithm that, given a value C, either proves that no feasible schedule of cost C exists, or else finds a schedule of cost at most C where each machine / is used for at most 27&quot;, time units. We also extend this result to a variant of the problem where, instead of a fixed processing time p,r there is a range of possible processing times for each machinejob pair, and the cost linearly increases as (he processing time decreases. We show that these results imply a polynomialtime 2approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomialtime algorithm that, given values M and 7&quot;, either proves that no schedule of mean job completion time M and makespan /&quot;exists, or else finds a schedule of mean job completion time at most M and makespan at most 2T.
Analysis of a local search heuristic for facility location problems
 IN PROCEEDINGS OF THE 9TH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1998
"... In this paper, we study approximation algorithms for several NPhard facility location problems. We prove that a simple local search heuristic yields polynomialtime constantfactor approximation bounds for the metric versions of the uncapacitated kmedian problem and the uncapacitated facility loca ..."
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Cited by 158 (4 self)
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In this paper, we study approximation algorithms for several NPhard facility location problems. We prove that a simple local search heuristic yields polynomialtime constantfactor approximation bounds for the metric versions of the uncapacitated kmedian problem and the uncapacitated facility location problem. (For the kmedian problem, our algorithms require a constantfactor blowup in the parameter k.) This local search heuristic was rst proposed several decades ago, and has been shown to exhibit good practical performance in empirical studies. We also extend the above results to obtain constantfactor approximation bounds for the metric versions of capacitated kmedian and facility location problems.
Clustering data streams: Theory and practice
 IEEE TKDE
, 2003
"... Abstract—The data stream model has recently attracted attention for its applicability to numerous types of data, including telephone records, Web documents, and clickstreams. For analysis of such data, the ability to process the data in a single pass, or a small number of passes, while using little ..."
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Cited by 154 (4 self)
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Abstract—The data stream model has recently attracted attention for its applicability to numerous types of data, including telephone records, Web documents, and clickstreams. For analysis of such data, the ability to process the data in a single pass, or a small number of passes, while using little memory, is crucial. We describe such a streaming algorithm that effectively clusters large data streams. We also provide empirical evidence of the algorithm’s performance on synthetic and real data streams. Index Terms—Clustering, data streams, approximation algorithms. 1
Provisioning a Virtual Private Network: A network design problem for multicommodity flow
 In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. Virtual private networks (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must ..."
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Cited by 111 (15 self)
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Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. Virtual private networks (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must be connected, bandwidth in the underlying network is reserved for communication within the group, forming a virtual “subnetwork.” Provisioning a virtual private network over a set of terminals gives rise to the following general network design problem. We have bounds on the cumulative amount of traffic each terminal can send and receive; we must choose a path for each pair of terminals, and a bandwidth allocation for each edge of the network, so that any traffic matrix consistent with the given upper bounds can be feasibly routed. Thus, we are seeking to design a network that can support a continuum of possible traffic scenarios. We provide optimal and approximate algorithms for several variants of this problem, depending on whether the traffic matrix is required to be symmetric, and on whether the designed network is required to be a tree (a natural constraint in a number of basic applications). We also establish a relation between this collection of network design problems and a variant of the facility location problem introduced by Karger and Minkoff; we extend their results by providing a stronger approximation algorithm for this latter problem. 1
The Online Median Problem
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... We introduce a natural variant of the (metric uncapacitated) kmedian problem that we call the online median problem. Whereas the kmedian problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities ar ..."
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Cited by 84 (2 self)
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We introduce a natural variant of the (metric uncapacitated) kmedian problem that we call the online median problem. Whereas the kmedian problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time; a facility cannot be moved once it is placed, and the total number of facilities to be placed, k, is not known in advance. The objective of an online median algorithm is to minimize the competitive ratio, that is, the worstcase ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a lineartime constantcompetitive algorithm for the online median problem. In addition, we present a related, though substantially simpler, lineartime constantfactor approximation algorithm for the (metric uncapacitated) facility location problem. The latter algorithm is similar in spirit to the recent primaldualbased facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time.
Hedging uncertainty: Approximation algorithms for stochastic optimization problems
 In Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization
, 2004
"... We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, ..."
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Cited by 82 (13 self)
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We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios. The approximation ratio of the stochastic variant of a typical problem is of the same order of magnitude as its deterministic counterpart. Furthermore, common techniques for designing approximation algorithms such as LP rounding, the primaldual method, and the greedy algorithm, can be carefully adapted to obtain these results. 1
Approximation Algorithms for Geometric Median Problems
, 1992
"... In this paper we present approximation algorithms for median problems in metric spaces and fixeddimensional Euclidean space. Our algorithms use a new method for transforming an optimal solution of the linear program relaxation of the smedian problem into a provably good integral solution. This ..."
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Cited by 81 (0 self)
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In this paper we present approximation algorithms for median problems in metric spaces and fixeddimensional Euclidean space. Our algorithms use a new method for transforming an optimal solution of the linear program relaxation of the smedian problem into a provably good integral solution. This transformation technique is fundamentally different from the methods of randomized and deterministic rounding [Rag, RaT] and the methods proposed in [LiV] in the following way: Previous techniques never set variables with zero values in the fractional solution to 1. This departure from previous methods is crucial for the success of our algorithms.