A scaleable projection-based branch-and-cut algorithm for the $p$-center problem

TL;DR

This paper introduces a scalable branch-and-cut algorithm for the p-center problem using a novel projection-based formulation, achieving strong LP bounds and competitive computational performance on large instances.

Contribution

It presents a new IP formulation derived by projection, enhanced with combinatorial cuts and lower bound information, improving solution efficiency for large-scale p-center problems.

Findings

Achieves LP-relaxation bounds comparable to the best in literature.

Effectively solves instances with over 700,000 customers and locations.

Competitive with existing set cover-based algorithms.

Abstract

The $p$-center problem (pCP) is a fundamental problem in location science, where we are given customer demand points and possible facility locations, and we want to choose $p$ of these locations to open a facility such that the maximum distance of any customer demand point to its closest open facility is minimized. State-of-the-art solution approaches of pCP use its connection to the set cover problem to solve pCP in an iterative fashion by repeatedly solving set cover problems. The classical textbook integer programming (IP) formulation of pCP is usually dismissed due to its size and bad linear programming (LP)-relaxation bounds. We present a novel solution approach that works on a new IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter LP-relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show that the LP-relaxation bound of our strengthened formulation has the same strength as the best known bound in literature, which is based on a semi-relaxation. Finally, we also present a computational study on instances from the literature with up to more than 700000 customers and locations. Our solution algorithm is competitive with highly sophisticated set-cover-based solution algorithms, which depend on various components and parameters.