Return of the primal-dual

Abstract

In this paper we present fast, distributed approximation algorithms for the metric facility location problem in the CONGEST model, where message sizes are bounded by O(log N) bits, N being the network size. We first show how to obtain a 7-approximation in O(log m + log n) rounds via the primal-dual method; here m is the number of facilities and n is the number of clients. Subsequently, we generalize this to a k-round algorithm, that for every constant k, yields an approximation factor of O(m2/√k ∙ n3/√k). These results answer a question posed by Moscibroda and Wattenhofer (PODC 2005). Our techniques are based on the primal-dual algorithm due to Jain and Vazirani (JACM 2001) and a rapid randomized sparsification of graphs due to Gfeller and Vicari (PODC 2007). These results complement the results of Moscibroda and Wattenhofer (PODC 2005) for non-metric facility location and extend the results of Gehweiler et al. (SPAA 2006) for uniform metric facility location.