Combinatorial problems such as combinatorial optimization and constraint satisfaction problems arise in decision-making across various fields of science and technology. In real-world applications, when multiple optimal or constraint-satisfying solutions exist, enumerating all these solutions -- rather than finding just one -- is often desirable, as it provides flexibility in decision-making. However, combinatorial problems and their enumeration versions pose significant computational challenges due to combinatorial explosion. To address these challenges, we propose enumeration algorithms for combinatorial optimization and constraint satisfaction problems using Ising machines. Ising machines are specialized devices designed to efficiently solve combinatorial problems. Typically, they sample low-cost solutions in a stochastic manner. Our enumeration algorithms repeatedly sample solutions to collect all desirable solutions. The crux of the proposed algorithms is their stopping criteria for sampling, which are derived based on probability theory. In particular, the proposed algorithms have theoretical guarantees that the failure probability of enumeration is bounded above by a user-specified value, provided that lower-cost solutions are sampled more frequently and equal-cost solutions are sampled with equal probability. Many physics-based Ising machines are expected to (approximately) satisfy these conditions. As a demonstration, we applied our algorithm using simulated annealing to maximum clique enumeration on random graphs. We found that our algorithm enumerates all maximum cliques in large dense graphs faster than a conventional branch-and-bound algorithm specially designed for maximum clique enumeration. This demonstrates the promising potential of our proposed approach.

翻译：组合优化与约束满足等组合问题广泛存在于科学技术各领域的决策过程中。在实际应用中，当存在多个最优解或满足约束的解时，枚举所有解（而非仅寻找单个解）通常更为可取，因其能为决策提供灵活性。然而，由于组合爆炸现象，组合问题及其枚举版本带来了显著的计算挑战。为应对这些挑战，本文提出基于伊辛机的组合优化与约束满足问题枚举算法。伊辛机是专为高效求解组合问题设计的特殊计算设备，通常以随机方式采样低成本解。本研究的枚举算法通过重复采样来收集所有目标解。算法的核心在于基于概率理论推导的采样停止准则。特别地，在满足“低成本解采样频率更高”与“等成本解等概率采样”的条件下，所提算法具有理论保证：枚举失败概率被控制在用户指定阈值以下。多数基于物理原理的伊辛机预期能（近似）满足这些条件。作为验证，我们通过模拟退火将算法应用于随机图的最大团枚举问题。实验表明，相较于专门为最大团枚举设计的传统分支定界算法，本算法在大型稠密图中能更快枚举所有最大团。这证明了所提方法具有广阔的应用前景。