New Relaxation-based Optimal Algorithms for the Solution of the Continuous and Discrete P -center Problems

The p-center problem (see, for example, [13]), also known as the minimax locationallocation problem, deals with the optimal location of emergency facilities. The locations of n demand points are given, and we need to locate p service facilities. The value of a candidate solution to the p-center problem is the maximum distance between a demand point to its nearest service facility. Our objective is to find the solution with the minimal value; we want to locate the service facilities so as to minimize the maximum distance between a demand point to its nearest service facility. It is assumed that all the facilities perform the same kind of service, and that the number of demand points that can get service from a given center is unlimited. Relaxation (in the context of this paper) [5, 13] is a method to optimally solve a large location problem by solving a succession of small sub-problems. It is an iterative algorithm which updates, at each step, bounds on the optimal solution, until the optimal solution is reached. This paper presents new relaxation algorithms for the p-center problem. Every step of a relaxation algorithm involves solving a p-center-like problem on a subset of the demand points. Our input is the subset and a value r, which is called the coverage distance. We need to answer: “Is there a solution to the sub-problem with value less than r?”. The new relaxation algorithms we describe try to reduce the number of iterations, or reduce the sizes of sub-problems, or reduce the values of the coverage distances (or a combination of these factors). There are two main variants of the p-center problem in the literature; they differ by the possible location of the service points. Many authors deal with the continuous problem in which the points to be located optimally can be anywhere in the plane, but another interesting problem is the discrete case where there is a finite set of potential points (xj , yj) out of which one wishes to find the points which fulfill the minimax condition. In some cases, weights wi are associated with the service points (ai, bi). Another classification of the problems is associated with the relevant metrics. In many cases, the distances between demand and service points are Euclidean (e.g. [9]). Also considered are problems where the distances are defined by minimal distances on a graph; this variant was first solved by Minieka [20].