Unsupervised Training of Diffusion Models for Feasible Solution Generation in Neural Combinatorial Optimization

TL;DR

This paper tackles neural combinatorial optimization for problems with two distinct item sets by introducing IC/DC, an unsupervised diffusion framework that directly generates feasible solutions without problem-specific supervision or search. It leverages a forward diffusion with a tailored noise transition and a reverse denoising process, augmented by feasibility-enforced generation and an alternating CLONING/IMPROVEMENT training loop guided by surrogate targets and reinforcement learning. Empirically, IC/DC achieves state-of-the-art results among learning-based methods on PMSP and ATSP, including 0% optimality gap on ATSP-20 and strong generalization to larger instances, while enabling faster inference with fewer diffusion steps. The approach broadens the applicability of diffusion-based methods to CO problems with complex feasibility constraints, reducing reliance on domain-specific heuristics and searches, though memory requirements of the bipartite GNN encoder remain a bottleneck for very large instances.

Abstract

Recent advancements in neural combinatorial optimization (NCO) methods have shown promising results in generating near-optimal solutions without the need for expert-crafted heuristics. However, high performance of these approaches often rely on problem-specific human-expertise-based search after generating candidate solutions, limiting their applicability to commonly solved CO problems such as Traveling Salesman Problem (TSP). In this paper, we present IC/DC, an unsupervised CO framework that directly trains a diffusion model from scratch. We train our model in a self-supervised way to minimize the cost of the solution while adhering to the problem-specific constraints. IC/DC is specialized in addressing CO problems involving two distinct sets of items, and it does not need problem-specific search processes to generate valid solutions. IC/DC employs a novel architecture capable of capturing the intricate relationships between items, and thereby enabling effective optimization in challenging CO scenarios. IC/DC achieves state-of-the-art performance relative to existing NCO methods on the Parallel Machine Scheduling Problem (PMSP) and Asymmetric Traveling Salesman Problem (ATSP).

Figures (6)

• Figure 1: The figure illustrates the CO problems we aim to address, and compares various learning-based approaches for CO problems. (Left) We focus on problems involving up to two distinct sets of items, which can be represented as a matrix. In ATSP, rows and columns represent cities (from/to), with each matrix element indicating distance. In PMSP, rows and columns represent machines and jobs, with each element representing processing time. (Center) The diagram highlights how these three approaches differ in generating solutions. Autoregressive methods generate solutions by iterating through items, while heatmap-based methods create a heatmap followed by a problem-specific heatmap search process. Our proposed IC/DC approach generates feasible solutions directly, using a training procedure that guides the diffusion model to satisfy the constraints.
• Figure 2: This figure illustrates our architecture and diffusion process. (Bottom-right) The problem encoder encodes a problem instance $c=(A, B, D)$ into a problem embedding $(A',B')$. (Bottom-left) The denoiser takes a problem embedding, noisy solution and a timestep embedding $(A', B', X_t,t)$ and outputs denoised solution $X_0$.
• Figure 3: (Left) shows the optimality gap among different methods in ATSP. Each of the methods used a unified sampling size of 128. The * indicates the training distribution (20 nodes). (Center) shows the optimality gap across sampling step size in PMSP (3 machines and 20 jobs). (Right) shows the inference time across sampling step size.
• Figure 4: Problem Instance $c$
• Figure 5: Solution Visualization