This paper presents a distributed O(1)-approximation algorithm, with expected-O(log log n) running time, in the CON GE ST model for the metric facility location problem on a size-n clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric facility location, extending the lower bound of Bȃdoiu et al. (ICALP 2005) that applies only to the special case of uniform facility opening costs. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an O(1)-ruling set of an appropriate spanning subgraph. Finally, we present a sub-logarithmic-round (in expectation) algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our algorithm accomplishes this by using a combination of randomized and deterministic sparsification.
For example, in a recent paper by Berns et al. REF a 2-ruling set is computed as a way of obtaining a O(1)-factor approximation to the metric facility location problem.
Super-Fast Distributed Algorithms for Metric Facility Location
ArXiv
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Computer Science