Primal-dual approximation algorithms for metric facility location and k-median problems

We present approximation algorithms for the metric uncapacitated facility location problem and the metric k-median problem achieving guarantees of 3 and 6 respectively. The distinguishing feature of our algorithms is their low running time: O(m log m) and O(m log m(L+log(n))) respectively, where n and m are the total number of vertices and edges in the underlying graph. The main algorithmic idea is a new extension of the primal-dual schema.

[1]  J. Stollsteimer.  A Working Model for Plant Numbers and Locations , 1963 .

[2]  Alfred A. Kuehn,et al.  A Heuristic Program for Locating Warehouses , 1963 .

[3]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[4]  Pierre Hansen,et al.  A Plant and Warehouse Location Problem , 1977 .

[5]  Reuven Bar-Yehuda,et al.  A Linear-Time Approximation Algorithm for the Weighted Vertex Cover Problem , 1981, J. Algorithms.

[6]  Dorit S. Hochbaum,et al.  Heuristics for the fixed cost median problem , 1982, Math. Program..

[7]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[8]  V. Rich.  Personal communication , 1989, Nature.

[9]  George L. Nemhauser,et al.  The uncapacitated facility location problem , 1990 .

[10]  Bhaba R. Sarker,et al.  Discrete location theory , 1991 .

[11]  R. Ravi,et al.  When trees collide: an approximation algorithm for the generalized Steiner problem on networks , 1991, STOC '91.

[12]  Said Salhi,et al.  Discrete Location Theory , 1991 .

[13]  Jeffrey Scott Vitter,et al.  Approximation Algorithms for Geometric Median Problems , 1992, Inf. Process. Lett..

[14]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[15]  Jeffrey Scott Vitter,et al.  e-approximations with minimum packing constraint violation (extended abstract) , 1992, STOC '92.

[16]  Judit Bar-Ilan,et al.  How to Allocate Network Centers , 1993, J. Algorithms.

[17]  Mihalis Yannakakis,et al.  Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees, with Applications to Matching and Set Cover , 1993, ICALP.

[18]  David P. Williamson,et al.  A primal-dual approximation algorithm for generalized steiner network problems , 1993, Comb..

[19]  Andrew V. Goldberg,et al.  Improved approximation algorithms for network design problems , 1994, SODA '94.

[20]  R. Ravi,et al.  When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks , 1995, SIAM J. Comput..

[21]  Samir Khuller,et al.  The Capacitated K-Center Problem (Extended Abstract) , 1996, ESA.

[22]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[23]  Éva Tardos,et al.  Approximation algorithms for facility location problems (extended abstract) , 1997, STOC '97.

[24]  An A Fabii,et al.  Improved Approximation Algorithms for Uncapacitated Facility Location , 1998 .

[25]  Rajmohan Rajaraman,et al.  Analysis of a local search heuristic for facility location problems , 2000, SODA '98.

[26]  Satish Rao,et al.  Approximation schemes for Euclidean k-medians and related problems , 1998, STOC '98.

[27]  Sudipto Guha,et al.  Rounding via Trees : Deterministic Approximation Algorithms forGroup , 1998 .

[28]  Samir Khuller,et al.  Greedy strikes back: improved facility location algorithms , 1998, SODA '98.

[29]  David P. Williamson,et al.  A primal-dual schema based approximation algorithm for the element connectivity problem , 2002, SODA '99.

[30]  David P. Williamson,et al.  Improved approximation algorithms for capacitated facility location problems , 1999, IPCO.

[31]  Fabián A. Chudak,et al.  Improved approximation algorithms for a capacitated facility location problem , 1999, SODA '99.

[32]  Sudipto Guha,et al.  A constant-factor approximation algorithm for the k-median problem (extended abstract) , 1999, STOC '99.

[33]  Vijay V. Vazirani,et al.  On the bidirected cut relaxation for the metric Steiner tree problem , 1999, SODA '99.

[34]  Paul S. Bradley,et al.  Mathematical Programming for Data Mining: Formulations and Challenges , 1999, INFORMS J. Comput..

[35]  Samir Khuller,et al.  The Capacitated K-Center Problem , 2000, SIAM J. Discret. Math..