Super-Fast Distributed Algorithms for Metric Facility Location

This paper explores the design of “super-fast” distributed algorithms in settings in which bandwidth constraints impose severe restrictions on the volume of information that can quickly reach an individual node. As a starting point of our exploration, we consider networks of diameter one (i.e., cliques) so as to focus on bandwidth constraints only and avoid penalties imposed by network distance between nodes. We assume the standard CONGEST model [17], which is a synchronous message-passing model in which each node in a size-n network can send a message of size O(log n) along each incident communication link in each round. By “super-fast” algorithms we mean algorithms whose running time is strictly sub-logarithmic. Our working hypothesis is that in low-diameter settings, where congestion rather than network distance is the main bottleneck, we should be able to design algorithms that are much faster than algorithms for “local” problems in high diameter settings. The focus of this paper is the distributed facility location problem which has been considered by several researchers [5,13,14,15] in low-diameter settings. We first describe the sequential version of the problem. The input to the facility location problem consists of a set of facilities F = {x1, x2, . . . , xm}, a set of clients C = {y1, y2, . . . , yn}, an opening cost fi associated with each facility xi, and a connection cost D(xi, yj) between each facility xi and client yj . The goal is to find a subset F ⊆ F of facilities to open so as to minimize the facility opening cost plus connection costs, i.e.,