Unsupervised Diffusion Solver for Combinatorial Optimization via Combinatorial Adjoint Matching

2026-05-29

AI Summary

This work addresses the challenge of applying diffusion-based methods to combinatorial optimization in the absence of labeled near-optimal solutions, a setting where existing supervised approaches fail. The authors propose the Combinatorial Adjoint Matching (CAM) framework, which extends adjoint trajectory optimization—previously limited to continuous domains—into discrete combinatorial spaces for the first time. By formulating the diffusion process as a stochastic control problem over a continuous-time Markov chain and leveraging discrete adjoint dynamics, CAM propagates low-variance, structured, trajectory-level optimization signals, enabling fully unsupervised training. Empirical results demonstrate that CAM significantly outperforms current unsupervised diffusion baselines across multiple combinatorial optimization tasks, achieving performance on par with strong supervised diffusion solvers and even classical optimizers.

Abstract

Diffusion-based neural solvers have shown strong promise for combinatorial optimization (CO), but existing methods typically rely on supervised training with large collections of near-optimal solutions. In this work, we extend adjoint-based trajectory optimization methods to discrete combinatorial domains. We formulate diffusion-based CO as a stochastic control problem over Continuous-Time Markov Chains and introduce discrete adjoint dynamics for propagating optimization signals through discrete generative trajectories. Building on this formulation, we propose Combinatorial Adjoint Matching (CAM), an unsupervised training framework for discrete diffusion solvers with structured and low-variance trajectory-level optimization signals. Empirically, CAM consistently outperforms existing unsupervised diffusion baselines and achieves performance competitive with strong supervised diffusion solvers and even traditional solvers across diverse combinatorial optimization problems. Our code is available at https://github.com/Shengyu-Feng/CAM.

Problem

• combinatorial optimization
• diffusion models
• unsupervised learning
• discrete adjoint dynamics
• stochastic control

Innovation

• Combinatorial Optimization
• Diffusion Models
• Adjoint Methods
• Unsupervised Learning
• Discrete Stochastic Control