Return of the primal-dual: distributed metric facilitylocation

In this paper we present fast, distributed approximation algorithms for the <i>metric facility location</i> problem in the <i>CONGEST</i> model, where message sizes are bounded by <i>O</i>(log <i>N</i>) bits, <i>N</i> being the network size. We first show how to obtain a 7-approximation in <i>O</i>(log <i>m</i> + log <i>n</i>) rounds via the primal-dual method; here <i>m</i> is the number of facilities and <i>n</i> is the number of clients. Subsequently, we generalize this to a <i>k</i>-round algorithm, that for every constant <i>k</i>, yields an approximation factor of <i>O</i>(<i>m</i><sup>2/√<i>k</i></sup> ∙ <i>n</i><sup>3/√<i>k</i></sup>). These results answer a question posed by Moscibroda and Wattenhofer (<i>PODC 2005</i>). Our techniques are based on the primal-dual algorithm due to Jain and Vazirani (<i>JACM 2001</i>) and a rapid randomized sparsification of graphs due to Gfeller and Vicari (<i>PODC 2007</i>). These results complement the results of Moscibroda and Wattenhofer (<i>PODC 2005</i>) for <i>non-metric</i> facility location and extend the results of Gehweiler et al. (<i>SPAA 2006</i>) for uniform metric facility location.

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