Super-Fast Distributed Algorithms for Metric Facility Location

TL;DR

This paper introduces a highly efficient distributed algorithm for metric facility location on clique networks, achieving constant-factor approximation in expected logarithmic double-logarithmic rounds, a significant advancement over previous methods.

Contribution

It presents the first sub-logarithmic-round distributed algorithm for metric facility location with non-uniform costs, utilizing a novel reduction to ruling set computation and new lower bounds.

Findings

Achieves O(1)-approximation in expected O(log log n) rounds

Introduces a new lower bound for metric facility location with non-uniform costs

Develops a sub-logarithmic-round algorithm for computing a 2-ruling set

Abstract

This paper presents a distributed O(1)-approximation algorithm, with expected-$O(\log \log n)$ running time, in the $\mathcal{CONGEST}$ 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 B\u{a}doiu 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.