Bi-Factor Approximation Algorithms for Hard Capacitated k-Median Problems

In the classical k-median problem the goal is to select a subset of at most k facilities in order to minimize the total cost of opened facilities and established connections between clients and opened facilities. We consider the capacitated version of the problem, where a single facility may only serve a limited number of clients. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard LP. It is well known that the standard LP has unbounded integrality gap if we only allow violating capacities by a factor smaller than 2, or if we only allow violating the number of facilities by a factor smaller than 2. In an earlier version of our work we showed that violating capacities by a factor of 2+ε is sufficient to obtain constant factor approximation of the connection cost. In this paper we substantially extend this result in the following two directions. First, we extend the 2+ε capacity violating algorithm to the more general k-facility location problem with uniform capacities, where opening facilities incurs a location specific opening cost. Second, we show that violating capacities by a slightly bigger factor of 3 + ε is sufficient to obtain constant factor approximation of the connection cost also in the case of the non-uniform hard capacitated k-median problem. Our algorithms first use the clustering of Charikar et al. to partition the facilities into sets of total fractional opening at least 1− 1/l for some fixed l. Then we exploit the technique of Levi, Shmoys, and Swamy developed for the capacitated facility location problem, which is to locally group the demand from clients to obtain a system of single node demand instances. Next, depending on the setting, we either use a dedicated routing tree on the demand nodes (for non-uniform opening cost), or we work with stars of facilities (for non-uniform capacities), to redistribute the demand that cannot be satissfied locally within the clusters. This research was supported by NCN 2012/07/N/ST6/03068 grant and by Warsaw Center of Mathematics and Computer Science from the KNOW grant of the Polish Ministry of Science and Higher Education. [email protected], [email protected] [email protected], [email protected]