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

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.