Combinatorial Optimization (CO) problems are fundamentally crucial in numerous practical applications across diverse industries, characterized by entailing enormous solution space and demanding time-sensitive response. Despite significant advancements made by recent neural solvers, their limited expressiveness does not conform well to the multi-modal nature of CO landscapes. While some research has pivoted towards diffusion models, they require simulating a Markov chain with many steps to produce a sample, which is time-consuming and does not meet the efficiency requirement of real applications, especially at scale. We propose DISCO, an efficient DIffusion Solver for Combinatorial Optimization problems that excels in both solution quality and inference speed. DISCO's efficacy is two-pronged: Firstly, it achieves rapid denoising of solutions through an analytically solvable form, allowing for direct sampling from the solution space with very few reverse-time steps, thereby drastically reducing inference time. Secondly, DISCO enhances solution quality by restricting the sampling space to a more constrained, meaningful domain guided by solution residues, while still preserving the inherent multi-modality of the output probabilistic distributions. DISCO achieves state-of-the-art results on very large Traveling Salesman Problems with 10000 nodes and challenging Maximal Independent Set benchmarks, with its per-instance denoising time up to 44.8 times faster. Through further combining a divide-and-conquer strategy, DISCO can be generalized to solve arbitrary-scale problem instances off the shelf, even outperforming models trained specifically on corresponding scales.

翻译：组合优化（CO）问题在众多行业的实际应用中具有根本重要性，其特点是解空间巨大且要求响应具有时效性。尽管近期神经求解器取得了显著进展，但其有限的表达能力难以很好地适应组合优化问题解空间的多模态特性。虽然已有研究转向扩散模型，但这些模型需要模拟包含多步的马尔可夫链来生成样本，这一过程耗时且无法满足实际应用（尤其是大规模场景）的效率要求。本文提出DISCO，一种用于组合优化问题的高效扩散求解器，在求解质量和推理速度方面均表现卓越。DISCO的有效性体现在两个方面：首先，它通过解析可解的形式实现解的快速去噪，允许以极少的反向时间步直接从解空间采样，从而大幅缩短推理时间。其次，DISCO通过将采样空间限制在由解残差引导的、约束性更强且更有意义的区域内来提升求解质量，同时仍保持输出概率分布固有的多模态特性。DISCO在具有10000个节点的超大规模旅行商问题及具有挑战性的最大独立集基准测试中取得了最先进的结果，其单实例去噪速度最高可提升44.8倍。通过进一步结合分治策略，DISCO能够直接推广至任意规模的问题实例求解，其表现甚至优于针对相应规模专门训练的模型。