Title: EDISCO: Equivariant Continuous-Time Categorical Diffusion for Geometric Combinatorial Optimization
Primary area: learning on graphs and other geometries & topologies
Keywords: Diffusion Models, Graph Neural Networks, Combinatorial Optimization, Equivariance, Geometric Deep Learning, Continuous-Time Diffusion

Abstract: Geometric combinatorial optimization problems, such as the Traveling Salesman Problem (TSP), possess inherent symmetries under rotations, translations, and reflections in Euclidean space. These transformations are denoted as E(2). However, existing neural network-based approaches, including recent diffusion-based solvers, fail to exploit these geometric features. This paper presents EDISCO, to the best of our knowledge, the first diffusion-based framework combining E(2)-equivariant graph neural networks with continuous-time categorical diffusion models for solving geometric combinatorial problems. This approach introduces an equivariant score network that respects geometric transformations while operating on discrete edge variables, together with a continuous-time categorical diffusion process that maintains E(2) symmetries throughout the forward and reverse processes. By incorporating geometric awareness directly into the diffusion process, EDISCO achieves notable improvements over the baseline. EDISCO reduces the state-of-the-art TSP optimality gaps on TSP-500 from 0.12\% to 0.08\%, TSP-1000 from 0.30\% to 0.22\%, and TSP-10000 from 2.68\% to 1.20\%. EDISCO demonstrates strong generalizability across problem sizes and also shows remarkable efficiency, requiring only 33\% to 50\% of the training data compared to competing diffusion methods across all problem scales.

TL;DR: EDISCO solves geometric problems via equivariant GNNs and continuous-time diffusion, enabling faster inference through advanced solvers while respecting geometric symmetries for SOTA results.