Cluster-based Message-Passing (CluMP) Optimization for Complex QUBO Problems

2026-06-13

AI Summary

This work addresses the challenge of highly frustrated energy landscapes in complex QUBO problems featuring conflicting constraints and long-range correlations, which often trap conventional local search algorithms in metastable states. To overcome this limitation, we propose CluMP, an algorithm that uniquely integrates belief propagation with non-local cluster updates. By leveraging message passing to identify low-frustration connected spin clusters, CluMP performs coordinated rearrangements within these clusters, thereby transcending the constraints of purely local dynamics. Empirical evaluations on diverse spin glass models demonstrate that CluMP significantly outperforms state-of-the-art local heuristics, achieving lower-energy configurations with fewer effective operations. This approach markedly enhances optimization efficiency and establishes a scalable new paradigm for tackling large-scale, sparse QUBO problems.

Abstract

Quadratic Unconstrained Boolean Optimization (QUBO) problems are widespread in both industrial applications and scientific studies. A QUBO problem corresponds to the optimization of a system of Ising spins defined on a generally sparse and heterogeneous graph. When the QUBO problem contains conflicting requests, the corresponding Ising system is frustrated, generating a complex energy landscape, which is hard to explore and optimize. Despite extensive algorithmic and hardware developments, finding low-energy configurations in these systems remains challenging (e.g., local-update heuristics typically become trapped in metastable states), especially when the (possibly frustrated) interactions generate extended correlated domains. We introduce CluMP (Cluster-based Message-Passing), an algorithm that performs collective updates on connected clusters of spins using information from Belief Propagation (BP). By controlling the amount of frustration within clusters, CluMP enables BP convergence on large subgraphs and proposes nonlocal rearrangements involving up to hundreds of spins in a single move. We benchmark CluMP against state-of-the-art local-update heuristics on spin-glass models defined on several graph topologies, including random regular graphs and lattice regular graphs in two and three dimensions. Cluster moves consistently bypass local trapping and reach lower energies with fewer effective operations than single-spin dynamics. These results demonstrate that frustration-tolerant cluster updates can be implemented efficiently on sparse graphs. The CluMP framework provides a scalable strategy for large-scale combinatorial optimization and inference problems, where exploiting medium- and long-range correlations is key to navigating complex energy landscapes.

Problem

• QUBO
• frustration
• energy landscape
• spin glass
• combinatorial optimization

Innovation

• Cluster-based Message-Passing
• QUBO optimization
• frustrated Ising systems
• belief propagation
• nonlocal spin updates