Quantum Optimization in Loc(Q)ation Science: QUBO Formulations, Benchmark Problems, and a Computational Study

TL;DR

This paper develops QUBO formulations for key location science problems, providing benchmarks for quantum algorithms, and conducts a computational study comparing quantum and classical methods with new warm-start strategies.

Contribution

It introduces novel QUBO models for location problems, derives a penalty bound, and performs a comprehensive computational comparison of quantum and classical algorithms.

Findings

QUBO formulations for location problems are effective benchmarks.

Quantum algorithms like QAOA show promising results on these benchmarks.

Warm-start strategies improve quantum algorithm performance.

Abstract

Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of $\mathbf{NP}$-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the $p$-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.