Hybrid Quantum-Classical Branch-and-Price Method for the Vertex Coloring Problem

Chiara Vercellino; M. Yassine Naghmouchi; Wesley Coelho; Giacomo Vitali; Alberto Scionti; Paolo Viviani; Olivier Terzo; Bartolomeo Montrucchio

arXiv · 2508.18887

Hybrid Quantum-Classical Branch-and-Price Method for the Vertex Coloring Problem

Chiara Vercellino·M. Yassine Naghmouchi·Wesley Coelho·Giacomo Vitali·Alberto Scionti·Paolo Viviani·Olivier Terzo·Bartolomeo Montrucchio

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Abstract

This paper introduces Quantum Classical Branch-and-Price (QCBP), a hybrid quantum-classical algorithm for the Vertex Coloring problem on neutral-atom Quantum Processing Units (QPUs). QCBP embeds quantum computation within the classical Branch-and-Price (BP) framework to address three bottlenecks in classical BP algorithms: the computational cost of Pricing Subproblems (PSPs), branching efficiency, and the quality of primal heuristics. It uses quantum-assisted Column Generation (CG) based on Quantum Adiabatic Algorithms (QAA) to sample high-quality maximum-weight independent sets (MWIS), reducing the need to repeatedly solve NP-hard PSPs. The adapted branching strategy leverages quantum-generated independent sets to explore fewer nodes, tighten lower bounds, and converge faster. A classical primal heuristic rapidly builds feasible solutions from quantum-generated sets, avoiding unnecessary quantum calls or additional Integer Linear Programming (ILP) solves. Compared with our prior Hybrid Column Generation (HCG) and Branch-and-Bound through maximal Independent Set (BBQ-mIS), QCBP improves both quantum-resource utilization and solution quality. Extensive experiments show QCBP significantly outperforms HCG and BBQ-mIS, reaching optimality on $\approx 98\%$ of benchmark instances. Preliminary validation on real neutral-atom hardware indicates robustness to quantum noise and hardware constraints, supporting practical applicability and scalability to larger graph instances. QCBP emerges as a viable hybrid method for combinatorial optimization with promising scalability on near-term quantum hardware.

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BibTeX RIS

Chiara Vercellino, M. Yassine Naghmouchi, Wesley Coelho, Giacomo Vitali, Alberto Scionti, Paolo Viviani, Olivier Terzo, Bartolomeo Montrucchio. 2025-08-26. Hybrid Quantum-Classical Branch-and-Price Method for the Vertex Coloring Problem. https://arxiv.org/abs/2508.18887

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