Super-Fast Distributed Algorithms for Metric Facility Location

This paper presents a distributed O(1)-approximation algorithm in the $\mathcal{CONGEST}$ model for the metric facility location problem on a size-n clique network that has an expected running time of O(loglogn ·log*n) rounds. 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. Since the facility location problem is specified by Ω(n2) bits of information, any fast solution in the $\mathcal{CONGEST}$ model must be truly distributed. Our paper makes three main technical contributions. First, we show a new lower bound for metric facility location. 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.

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