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214
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 769 (5 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
Primaldual approximation algorithms for metric facility location and kmedian problems
 Journal of the ACM
, 1999
"... ..."
Local search heuristics for kmedian and facility location problems
, 2001
"... ÔÖÓ��ÙÖ�ØÓØ���ÐÓ��ÐÓÔØ�ÑÙÑ�ÓÖ�Ñ����ÒÛ � Ö�Ø�ÓÓ��ÐÓ�ÐÐÝÓÔØ�ÑÙÑ×ÓÐÙØ�ÓÒÓ�Ø��Ò��Ù×�Ò�Ø�� × ÐÓ�Ð�ØÝ��ÔÓ��ÐÓ�Ð×��Ö�ÔÖÓ��ÙÖ��×Ø��Ñ�Ü�ÑÙÑ �Ñ����Ò�Ò����Ð�ØÝÐÓ�Ø�ÓÒÔÖÓ�Ð�Ñ×Ï���¬Ò�Ø� � ÁÒØ��×Ô�Ô�ÖÛ��Ò�ÐÝÞ�ÐÓ�Ð×��Ö���ÙÖ�×Ø�×�ÓÖØ�� ×�ÓÛØ��ØÐÓ�Ð×��Ö�Û�Ø�×Û�Ô×��×�ÐÓ�Ð�ØÝ��ÔÓ � ×�ÑÙÐØ�Ò�ÓÙ×ÐÝØ��ÒØ��ÐÓ�Ð�ØÝ��ÔÓ�Ø�� ..."
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Cited by 295 (10 self)
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ÔÖÓ��ÙÖ�ØÓØ���ÐÓ��ÐÓÔØ�ÑÙÑ�ÓÖ�Ñ����ÒÛ � Ö�Ø�ÓÓ��ÐÓ�ÐÐÝÓÔØ�ÑÙÑ×ÓÐÙØ�ÓÒÓ�Ø��Ò��Ù×�Ò�Ø�� × ÐÓ�Ð�ØÝ��ÔÓ��ÐÓ�Ð×��Ö�ÔÖÓ��ÙÖ��×Ø��Ñ�Ü�ÑÙÑ �Ñ����Ò�Ò����Ð�ØÝÐÓ�Ø�ÓÒÔÖÓ�Ð�Ñ×Ï���¬Ò�Ø� � ÁÒØ��×Ô�Ô�ÖÛ��Ò�ÐÝÞ�ÐÓ�Ð×��Ö���ÙÖ�×Ø�×�ÓÖØ�� ×�ÓÛØ��ØÐÓ�Ð×��Ö�Û�Ø�×Û�Ô×��×�ÐÓ�Ð�ØÝ��ÔÓ � ×�ÑÙÐØ�Ò�ÓÙ×ÐÝØ��ÒØ��ÐÓ�Ð�ØÝ��ÔÓ�Ø��ÐÓ�Ð×��Ö � �Ü�ØÐÝ�Ï��ÒÛ�Ô�ÖÑ�ØÔ���Ð�Ø��×ØÓ��×Û�ÔÔ�� �ÑÔÖÓÚ�×Ø��ÔÖ�Ú�ÓÙ×�ÒÓÛÒ��ÔÔÖÓÜ�Ñ�Ø�ÓÒ�ÓÖØ�� × ÔÖÓ�Ð�Ñ�ÓÖÍÒ�Ô��Ø�Ø�����Ð�ØÝÐÓ�Ø�ÓÒÛ�×�ÓÛ ÔÖÓ��ÙÖ��×�Ü�ØÐÝ Ó�ÐÓ�Ð×��Ö��ÓÖ�Ñ����ÒØ��ØÔÖÓÚ���×��ÓÙÒ�� � Ô�Ö�ÓÖÑ�Ò��Ù�Ö�ÒØ��Û�Ø�ÓÒÐÝ�Ñ����Ò×Ì��×�Ð×Ó �ÔÌ��×�×Ø��¬Ö×Ø�Ò�ÐÝ×�× ×Û�ÔÔ�Ò�����Ð�ØÝ��×�ÐÓ�Ð�ØÝ��ÔÓ��Ü�ØÐÝÌ�� × �ÑÔÖÓÚ�×Ø����ÓÙÒ�Ó�ÃÓÖÙÔÓÐÙ�Ø�ÐÏ��Ð×ÓÓÒ ×���Ö��Ô��Ø�Ø�����Ð�ØÝÐÓ�Ø�ÓÒÔÖÓ�Ð�ÑÛ��Ö��� � Ø��ØÐÓ�Ð×��Ö�Û���Ô�ÖÑ�Ø×����Ò��ÖÓÔÔ�Ò��Ò� Ø�ÔÐ�ÓÔ��×Ó�����Ð�ØÝ�ÓÖØ��×ÔÖÓ�Ð�ÑÛ��ÒØÖÓ�Ù � ���Ð�ØÝ��×��Ô��ØÝ�Ò�Û��Ö��ÐÐÓÛ��ØÓÓÔ�ÒÑÙÐ ÐÓ�Ð×��Ö�Û���Ô�ÖÑ�Ø×Ø��×Ò�ÛÓÔ�Ö�Ø�ÓÒ��×�ÐÓ ���Ð�ØÝ�Ò��ÖÓÔ×Þ�ÖÓÓÖÑÓÖ����Ð�Ø��×Ï�ÔÖÓÚ�Ø��Ø �Ò�ÛÓÔ�Ö�Ø�ÓÒÛ���ÓÔ�Ò×ÓÒ�ÓÖÑÓÖ�ÓÔ��×Ó� � �Ð�ØÝ��Ô��ØÛ��Ò�Ò�� ÝÈ�ÖØ��ÐÐÝ×ÙÔÔÓÖØ���Ý���ÐÐÓÛ×��Ô�ÖÓÑÁÒ�Ó×Ý×Ì� � Ê�×��Ö�Ä� � ÒÓÐÓ���×ÄØ���Ò��ÐÓÖ � ÞËÙÔÔÓÖØ���Ý�ÊÇ������� � £È�ÖØ��ÐÐÝ×ÙÔÔÓÖØ���Ý���ÐÐÓÛ×��Ô�ÖÓÑÁ�ÅÁÒ���
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 251 (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 228 (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....
The Budgeted Maximum Coverage Problem
, 1997
"... The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L, find a subset of S 0 ` S such that the total cost of sets in S 0 does not exceed L, and the total weight of elements covered by S 0 is maxim ..."
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Cited by 184 (7 self)
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The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L, find a subset of S 0 ` S such that the total cost of sets in S 0 does not exceed L, and the total weight of elements covered by S 0 is maximized. This problem is NPhard. For the special case of this problem, where each set has unit cost, a (1 \Gamma 1 e )approximation is known. Yet, no approximation results are known for the general cost version. The contribution of this paper is a (1 \Gamma 1 e )approximation algorithm for the budgeted maximum coverage problem. We also argue that this approximation factor is the best possible, unless NP ` DT IME(n log log n ). 1 Introduction The budgeted maximum coverage problem is defined as follows. A collection of sets S = fS 1 ; S 2 ; : : : ; Sm g with associated costs fc i g m i=1 is defined over a domain of elements X = fx 1 ; x 2 ; : : : ; x n g with associated weights fw i ...
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 159 (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 149 (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
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 147 (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.
Improved Approximation Algorithms for Metric Facility Location Problems
 In Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization
, 2002
"... In this paper we present a 1.52approximation algorithm for the metric uncapacitated facility location problem, and a 2approximation algorithm for the metric capacitated facility location problem with soft capacities. Both these algorithms improve the best previously known approximation factor for ..."
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Cited by 144 (11 self)
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In this paper we present a 1.52approximation algorithm for the metric uncapacitated facility location problem, and a 2approximation algorithm for the metric capacitated facility location problem with soft capacities. Both these algorithms improve the best previously known approximation factor for the corresponding problem, and our softcapacitated facility location algorithm achieves the integrality gap of the standard LP relaxation of the problem. Furthermore, we will show, using a result of Thorup, that our algorithms can be implemented in quasilinear time.