On the P-coverage Problem on the Real Line

In this paper we consider the p-coverage problem on the real line. We first give a detailed description of an algorithm to solve the coverage problem without the upper bound p on the number of open facilities. Then we analyze how the structure of the optimal solution changes if the setup costs of the facilities are all decreased by the same amount. This result is used to develop a parametric approach to the p-coverage problem which runs in O(pnlogn) time, n being the number of clients. OR/MS subject classification: Analysis of algorithms, computational complexity: parametric application of dynamic programming; Dynamic programming/optimal control, applications: parametric approach to p-coverage problem on the real line; Facilities/equipment planning, location, discrete: p-coverage problem on the real line Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. 2 Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands; currently on leave at the Operations Research Center, Massachusetts Institute of Technology, Cambridge MA; financial support of the Netherlands Organization for Scientific Research (NWO) is gratefully acknowledged. O. Introduction In Hassin and Tamir (1990) recent results in dynamic programming are used to improve the complexity bounds of several median and coverage location models on the real line. A general model that unifies some of the classical problems in location theory is shown to be solvable in O(n2) time, where n is the number of clients. One of the special cases of this general model is the p coverage problem, where as usual p refers to an upper bound on the number of open facilities. Hassin and Tamir show that if this upper bound is ignored (or redundant), the resulting problem is solvable in O(nlogn) time. Because the p-coverage problem can be formulated as a 0/1 linear program with a totally unimodular restriction matrix, an optimal solution to the Lagrangean dual problem that results from relaxing the upper bound constraint yields an optimal solution to the p-coverage problem. To find that optimal solution a parametric method due to Megiddo (1979) is used, resulting in an O(n21og2n) algorithm. In this paper we present an algorithm that solves the p-coverage problem on the real line in O(pnlogn) time. After describing the problem in Section 1, we give a detailed description of an algorithm to solve the coverage problem without the upper bound on the number of open facilities (Section 2). This algorithm takes O(nlogn) time. We consider the actual p-coverage problem in Section 3. First we analyze how the structure of the optimal solution changes if the setup costs of the facilities are all decreased by the same amount. Then this result is used to develop a parametric approach to the p-coverage problem which runs in O(pnlogn) time. In Section 4 we describe how the algorithm should be modified to obtain an O(pn) algorithm for two special cases of the p-coverage problem. Section 5 contains some concluding remarks. 1. Problem description We consider the p-coverage problem in which n distinct points, v1 to vn, are located on a line. These points represent both the set of clients and the set of potential facility sites. To facilitate the exposition, we assume that the points are numbered from left to right, i.e., j<m if and only if vj is located to the left of vm. Let d(vi, vj) denote the distance between the points vi and vj. With the client at vi we associate the radius ri, which has the interpretation that this client can only be served by facilities at vertices vj for which d(vi, vj) <ri. We will say that a client is covered by a subset S of facilities, if S contains at least one facility that can serve that client.