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IDEQ: an improved diffusion model for the TSP

Mickael Basson, Philippe Preux

TL;DR

IDEQ addresses solving the TSP with neural diffusion models by integrating two innovations: enforcing the Hamiltonian structure during inference via a reconstruction operator and refining via 2-opt, and fine-tuning through equivalence-class sampling over 2-opt. These approaches guide the diffusion process toward high-quality Hamiltonian tours and more robust training. The method achieves state-of-the-art results on TSPlib and competitive performance against LKH3 on large instances, with notably lower variability. It demonstrates that reusing curriculum checkpoints and combining constructive post-processing with diffusion could generalize to other combinatorial problems and inspires future extensions for faster, higher-quality neural TSP solvers.

Abstract

We investigate diffusion models to solve the Traveling Salesman Problem. Building on the recent DIFUSCO and T2TCO approaches, we propose IDEQ. IDEQ improves the quality of the solutions by leveraging the constrained structure of the state space of the TSP. Another key component of IDEQ consists in replacing the last stages of DIFUSCO curriculum learning by considering a uniform distribution over the Hamiltonian tours whose orbits by the 2-opt operator converge to the optimal solution as the training objective. Our experiments show that IDEQ improves the state of the art for such neural network based techniques on synthetic instances. More importantly, our experiments show that IDEQ performs very well on the instances of the TSPlib, a reference benchmark in the TSP community: it closely matches the performance of the best heuristics, LKH3, being even able to obtain better solutions than LKH3 on 2 instances of the TSPlib defined on 1577 and 3795 cities. IDEQ obtains 0.3% optimality gap on TSP instances made of 500 cities, and 0.5% on TSP instances with 1000 cities. This sets a new SOTA for neural based methods solving the TSP. Moreover, IDEQ exhibits a lower variance and better scales-up with the number of cities with regards to DIFUSCO and T2TCO.

IDEQ: an improved diffusion model for the TSP

TL;DR

IDEQ addresses solving the TSP with neural diffusion models by integrating two innovations: enforcing the Hamiltonian structure during inference via a reconstruction operator and refining via 2-opt, and fine-tuning through equivalence-class sampling over 2-opt. These approaches guide the diffusion process toward high-quality Hamiltonian tours and more robust training. The method achieves state-of-the-art results on TSPlib and competitive performance against LKH3 on large instances, with notably lower variability. It demonstrates that reusing curriculum checkpoints and combining constructive post-processing with diffusion could generalize to other combinatorial problems and inspires future extensions for faster, higher-quality neural TSP solvers.

Abstract

We investigate diffusion models to solve the Traveling Salesman Problem. Building on the recent DIFUSCO and T2TCO approaches, we propose IDEQ. IDEQ improves the quality of the solutions by leveraging the constrained structure of the state space of the TSP. Another key component of IDEQ consists in replacing the last stages of DIFUSCO curriculum learning by considering a uniform distribution over the Hamiltonian tours whose orbits by the 2-opt operator converge to the optimal solution as the training objective. Our experiments show that IDEQ improves the state of the art for such neural network based techniques on synthetic instances. More importantly, our experiments show that IDEQ performs very well on the instances of the TSPlib, a reference benchmark in the TSP community: it closely matches the performance of the best heuristics, LKH3, being even able to obtain better solutions than LKH3 on 2 instances of the TSPlib defined on 1577 and 3795 cities. IDEQ obtains 0.3% optimality gap on TSP instances made of 500 cities, and 0.5% on TSP instances with 1000 cities. This sets a new SOTA for neural based methods solving the TSP. Moreover, IDEQ exhibits a lower variance and better scales-up with the number of cities with regards to DIFUSCO and T2TCO.

Paper Structure

This paper contains 18 sections, 2 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and related works
  3. About the TSP
  4. Denoising diffusion models
  5. Diffusion models for combinatorial optimization
  6. Other constructive neural solvers for combinatorial optimization
  7. Methods
  8. Background on discrete diffusion models for combinatorial optimization
  9. Key ingredient 1: leveraging the constrained structure of the solution space to improve inference-time solution generation
  10. Key ingredient 2: leveraging equivalence class over target distribution to fine-tune diffusion checkpoints
  11. IDEQ
  12. Experimental study
  13. Experimental setup
  14. Experimental methodology
  15. Experimental results
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Illustration of 2-change and 2-opt operators on a Hamiltonian tour.
  • Figure 2: Distribution of optimality gaps measured on 32 repetitions of the first 64 TSP-1000 instances from the test set. IDEQ in blue and T2TCO in orange. Bars represent the empirical data ditribution and the line is the Gaussian kernel density estimate.