The Complete Vertex p-Center Problem

TL;DR

This paper introduces two efficient algorithms for solving the complete vertex p-center problem, enabling rapid computation of optimal facility locations across all p values to aid spatial planning decisions.

Contribution

It presents novel algorithms that significantly speed up solving the complete p-center problem, including a trimming variable approach and a set covering conversion.

Findings

Set covering method reduces computation time by orders of magnitude.

Algorithms provide exact solutions for all p values efficiently.

Trade-off curve helps decision-makers visualize coverage options.

Abstract

The vertex p-center problem consists of locating p facilities among a set of M potential sites such that the maximum distance from any demand to its closest located facility is minimized. The complete vertex p-center problem solves the p-center problem for all p from 1 to the total number of sites, resulting in a multi-objective trade-off curve between the number of facilities and the service distance required to achieve full coverage. This trade-off provides a reference to planners and decision-makers, enabling them to easily visualize the consequences of choosing different coverage design criteria for the given spatial configuration of the problem. We present two fast algorithms for solving the complete p-center problem, one using the classical formulation but trimming variables while still maintaining optimality, the other converting the problem to a location set covering problem and solving for all distances in the distance matrix. We also discuss scenarios where it makes sense to solve the problem via brute-force enumeration. All methods result in significant speed-ups, with the set covering method reducing computation times by many orders of magnitude.