The FastMap Pipeline for Facility Location Problems

Abstract

Facility Location Problems (FLPs) involve the placement of facilities in a shared environment for serving multiple customers while minimizing transportation and other costs. FLPs defined on graphs are very general and broadly applicable. Two such fundamental FLPs are the Vertex K-Center (VKC) and the Vertex K-Median (VKM) problems. Although both these problems are NP-hard, many heuristic and approximation algorithms have been developed for solving them in practice. However, state-of-the-art heuristic algorithms require the input graph G to be complete, in which the edge joining two vertices is also the shortest path between them. When G doesn’t satisfy this property, these heuristic algorithms have to be invoked only after computing the metric closure of G, which in turn requires the computation of all-pairs shortest-path (APSP) distances. Existing APSP algorithms, such as the Floyd-Warshall algorithm, have a poor time complexity, making APSP computations a bottleneck for deploying the heuristic algorithms on large VKC and VKM instances. To remedy this, we propose the use of a novel algorithmic pipeline based on a graph embedding algorithm called FastMap. FastMap is a near-linear-time algorithm that embeds the vertices of G in a Euclidean space while approximately preserving the shortest-path distances as Euclidean distances for all pairs of vertices. The FastMap embedding can be used to circumvent the barrier of APSP computations, creating a very efficient pipeline for solving FLPs. On the empirical front, we provide test results that demonstrate the efficiency and effectiveness of our novel approach.