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：组合优化问题在众多行业的实际应用中具有根本重要性，其特点在于解空间巨大且通常需要满足时效性响应。尽管近期基于神经网络的求解器取得了显著进展，但其有限的表达能力难以适应组合优化问题解空间固有的多模态特性。虽然部分研究已转向扩散模型，但这些方法需要模拟包含多步的马尔可夫链来生成样本，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——一种针对组合优化问题的高效扩散求解器，在求解质量与推理速度方面均表现卓越。DISCO的有效性体现在双重机制：首先，通过解析可解形式实现解的快速去噪，仅需极少反向时间步即可直接从解空间采样，从而大幅降低推理时间；其次，在解残差引导下将采样空间限制于更具约束性的有效区域，同时保持输出概率分布固有的多模态特性，从而提升求解质量。DISCO在万节点规模的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成果，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，其性能甚至优于针对特定规模专门训练的模型。