Compact MILP formulations for the $p$-center problem

TL;DR

This paper introduces two new compact integer formulations for the p-center problem, improving computational efficiency and relaxation bounds, supported by an algorithm for strong bounds and extensive computational comparisons.

Contribution

The paper presents two novel compact MILP formulations for the p-center problem, improving upon previous models and introducing an algorithm for enhanced bounds and reduced problem size.

Findings

The first formulation significantly reduces the number of constraints.

The second formulation has fewer variables but a weaker relaxation bound.

The proposed algorithm improves bounds and reduces formulation size.

Abstract

The p-center problem consists in selecting p centers among M to cover N clients, such that the maximal distance between a client and its closest selected center is minimized. For this problem we propose two new and compact integer formulations. Our first formulation is an improvement of a previous formulation. It significantly decreases the number of constraints while preserving the optimal value of the linear relaxation. Our second formulation contains less variables and constraints but it has a weaker linear relaxation bound. We besides introduce an algorithm which enables us to compute strong bounds and significantly reduce the size of our formulations. Finally, the efficiency of the algorithm and the proposed formulations are compared in terms of quality of the linear relaxation and computation time over instances from OR-Library.