Investigating mixed-integer programming approaches for the $p$-$α$-closest-center problem
2026-03-13 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study a generalization of the facility location problem called the p-α-closest-center problem, where the goal is to open p facilities to minimize the sum of distances to each customer's α closest open facilities. They provide four mathematical models and improve them using inequalities and theoretical insights to better estimate the solution. The authors then develop an exact algorithm based on branch-and-cut techniques, enhanced with heuristics and fixes, to solve the problem efficiently. Their computational tests show that their method can find optimal solutions for many instances where previous methods could not.
facility location problemp-center problemp-second-center problemmixed-integer programmingbranch-and-cut algorithmlinear programming relaxationvalid inequalitiesheuristicsoptimizationpolyhedral study
Authors
Elisabeth Gaar, Sara Joosten, Markus Sinnl
Abstract
In this work, we introduce and study the $p$-$α$-closest-center problem ($pα$CCP), which generalizes the $p$-second-center problem, a recently emerged variant of the classical $p$-center problem. In the $pα$CCP, we are given sets of customers and potential facility locations, distances between each customer and potential facility location as well as two integers $p$ and $α$. The goal is to open facilities at $p$ of the potential facility locations, such that the maximum $α$-distance between each customer and the open facilities is minimized. The $α$-distance of a customer is defined as the sum of distances from the customer to its $α$ closest open facilities. If $α$ is one, the $pα$CCP is the $p$-center problem, and for $α$ being two, the $p$-second-center problem is obtained, for which the only existing algorithm in literature is a variable neighborhood search (VNS). We present four mixed-integer programming (MIP) formulations for the $pα$CCP, strengthen them by adding valid and optimality-preserving inequalities and conduct a polyhedral study to prove relationships between their linear programming relaxations. Moreover, we present iterative procedures for lifting some valid inequalities to improve initial lower bounds on the optimal objective function value of the $pα$CCP and characterize the best lower bounds obtainable by this iterative lifting approach. Based on our theoretical findings, we develop a branch-and-cut algorithm (B&C) to solve the $pα$CCP exactly. We improve its performance by a starting and a primal heuristic, variable fixings and separating inequalities. In our computational study, we investigate the effect of the various ingredients of our B&C on benchmark instances from related literature. Our B&C is able to prove optimality for 17 of the 40 instances from the work on the VNS heuristic.