Solving Facility Location Problems via FastMap and Locality Sensitive Hashing

Abstract

Facility Location Problems (FLPs) arise while serving multiple customers in a shared environment, minimizing transportation and other costs. Hence, they involve the optimal placement of facilities. They are defined on graphs as well as in Euclidean spaces with or without obstacles; and they are typically NP-hard to solve optimally. There are many heuristic algorithms tailored to different kinds of FLPs. However, FLPs defined in Euclidean spaces without obstacles are the most amenable to efficient and effective heuristic algorithms. This motivates the idea of quickly reformulating FLPs on graphs and in Euclidean spaces with obstacles to FLPs in Euclidean spaces without obstacles. Towards this end, we propose a new approach that uses FastMap and Locality Sensitive Hashing. FastMap is a near-linear-time algorithm that embeds the vertices of a graph in a Euclidean space while approximately preserving graph-based distances as Euclidean distances for all pairs of vertices. Through extensive experiments, we show that our approach significantly outperforms other state-of-the-art competing algorithms on a variety of FLPs: the Multi-Agent Meeting, Vertex K-Median (VKM), Weighted VKM, and the Capacitated VKM problems.