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

TL;DR

This work develops QUBO formulations for pivotal location-science problems (including $p$-Median, $p$-Center, FCFLP, GAP, DOMP, and DMPFLP) and proves a tight penalty-parameter bound that preserves equivalence to the underlying integer programs. It investigates quantum optimization via QAOA and WS-QAOA, augmented by two LP-based warm-start strategies, using noiseless simulations to reveal algorithmic strengths and limitations. The study provides extensive numerical results for small instances, showing that LP-based warm starts can significantly improve WS-QAOA performance and that the QUBO formulations can outperform classical heuristics in certain settings. Together, these contributions offer practical, benchmark-ready quantum-ready formulations and insights into how polyhedral structure and warm-starts influence quantum optimization on current hardware trajectories.

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.

Figures (5)

• Figure 1: $p$-Median Problem ($n = 3$ and $p = 1$)
• Figure 2: $p$-Median Problem ($n = 3$ and $p = 2$)
• Figure 3: $p$-Median Problem ($n = 4$ and $p = 2$)
• Figure 4: FCFLP (aggregated formulation and $n = 3$)
• Figure 5: FCFLP (disaggregated formulation and $n = 3$)

Theorems & Definitions (33)

• Proposition 1: QUBO:Bochkarev, Remark 1
• Theorem 2
• proof
• Corollary 1
• proof
• Example 1
• Lemma 1
• proof
• Corollary 2
• proof