An approximation algorithm for Uniform Capacitated k-Median problem with 1 + ε capacity violation

We study the Capacitated k-Median problem, for which all the known constant factor approximation algorithms violate either the number of facilities or the capacities. While the standard LP-relaxation can only be used for algorithms violating one of the two by a factor of at least two, Li [10, 11] gave algorithms violating the number of facilities by a factor of $$1+\epsilon $$ exploring properties of extended relaxations. In this paper we develop a constant factor approximation algorithm for hard Uniform Capacitated k-Median violating only the capacities by a factor of $$1\,+\,\epsilon $$. The algorithm is based on a configuration LP. Unlike in the algorithms violating the number of facilities, we cannot simply open extra few facilities at selected locations. Instead, our algorithm decides about the facility openings in a carefully designed dependent rounding process.

[1]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

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

[3]  Shi Li.  On Uniform Capacitated k-Median Beyond the Natural LP Relaxation , 2015, SODA.

[4]  Maxim Sviridenko.  An Improved Approximation Algorithm for the Metric Uncapacitated Facility Location Problem , 2002, IPCO.

[5]  Aravind Srinivasan,et al.  An Improved Approximation for k-Median and Positive Correlation in Budgeted Optimization , 2014, SODA.

[6]  Mohit Singh,et al.  LP-Based Algorithms for Capacitated Facility Location , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[7]  Yuval Rabani,et al.  Approximating k-median with non-uniform capacities , 2005, SODA '05.

[8]  Shi Li,et al.  On Uniform Capacitated k-Median Beyond the Natural LP Relaxation , 2014, SODA.

[9]  Shi Li.  Approximating capacitated k-median with (1 + ∊)k open facilities , 2014, ACM-SIAM Symposium on Discrete Algorithms.

[10]  Karen Aardal,et al.  Approximation algorithms for hard capacitated k-facility location problems , 2013, Eur. J. Oper. Res..

[11]  Kamesh Munagala,et al.  Local Search Heuristics for k-Median and Facility Location Problems , 2004, SIAM J. Comput..

[12]  Jaroslaw Byrka,et al.  Bi-Factor Approximation Algorithms for Hard Capacitated k-Median Problems , 2013, SODA.

[13]  Shi Li,et al.  Approximating k-median via pseudo-approximation , 2012, STOC '13.

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

[15]  Shi Li,et al.  Approximating capacitated k-median with (1 + ∊)k open facilities , 2014, SODA.

[16]  Rajiv Gandhi,et al.  Dependent rounding and its applications to approximation algorithms , 2006, JACM.