🤖 AI Summary
In real-world scenarios, enumerating all optimal or feasible solutions to combinatorial optimization or constraint satisfaction problems is often required for informed decision-making; however, deterministic algorithms suffer from combinatorial explosion and poor scalability. This paper proposes a sampling-based solution enumeration framework tailored for physical Ising machines: the problem is formulated as an Ising model, and controlled probabilistic sampling is employed to explore the energy landscape. Crucially, we introduce the first theoretically grounded stopping criterion with statistical guarantees—ensuring the probability of incomplete enumeration remains below a user-specified threshold. By embracing the intrinsic stochasticity of Ising hardware, our approach departs from deterministic paradigms. Evaluated on maximum clique enumeration, it significantly outperforms specialized branch-and-bound algorithms, enabling efficient and complete enumeration of all maximum cliques in large-scale dense graphs.
📝 Abstract
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 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 by exploring the energy landscape of an Ising model. Ising machines typically sample lower-energy solutions with higher probability. Our enumeration algorithms repeatedly perform such sampling to collect all desirable solutions. The crux of the proposed algorithms lies in their stopping criteria for sampling-based energy landscape exploration, which are derived from 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 specifically designed for maximum clique enumeration. These findings underscore the effectiveness and potential of our proposed approach.