Transport, Don't Generate: Deterministic Geometric Flows for Combinatorial Optimization

TL;DR

This paper introduces CycFlow, a deterministic geometric flow method for solving the Euclidean TSP that significantly speeds up computation by replacing stochastic diffusion models with continuous point transport, maintaining competitive accuracy.

Contribution

CycFlow presents a novel deterministic point transport framework that replaces stochastic diffusion models for TSP, achieving faster solutions with comparable quality.

Findings

Speeds up TSP solving by up to 1000x compared to diffusion models.

Maintains competitive optimality gaps.

Uses linear coordinate dynamics instead of quadratic edge scoring.

Abstract

Recent advances in Neural Combinatorial Optimization (NCO) have been dominated by diffusion models that treat the Euclidean Traveling Salesman Problem (TSP) as a stochastic $N \times N$ heatmap generation task. In this paper, we propose CycFlow, a framework that replaces iterative edge denoising with deterministic point transport. CycFlow learns an instance-conditioned vector field that continuously transports input 2D coordinates to a canonical circular arrangement, where the optimal tour is recovered from this $2N$ dimensional representation via angular sorting. By leveraging data-dependent flow matching, we bypass the quadratic bottleneck of edge scoring in favor of linear coordinate dynamics. This paradigm shift accelerates solving speed by up to three orders of magnitude compared to state-of-the-art diffusion baselines, while maintaining competitive optimality gaps.