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

TL;DR

CycFlow tackles Euclidean TSP by reframing it as a deterministic geometric transport problem. It learns a conditional velocity field that deterministically maps the input point set $x_0$ to a circle-embedded target $x_1$, enabling the tour to be recovered by angular sorting and a light refinement. The core contributions are data-dependent geometric couplings that construct $x_1$, a flow-matching objective for a velocity field, and a Canonicalize-Process-Restore architecture that preserves geometric symmetries and enables scalable, permutation-equivariant inference. Empirically, CycFlow achieves sub-second inference up to $N=1000$ with competitive gaps, delivering 2–3 orders of magnitude speedups over diffusion-based solvers while maintaining practical solution quality. This work shows that aligning generative inference with the problem's Euclidean geometry can unlock real-time, scalable combinatorial optimization.

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.

Figures (4)

• Figure 1: Comparison of the prevailing NCO paradigm (Left), which views TSP as stochastic heatmap edge denoising, versus CycFlow (Right), which treats TSP as a deterministic geometric flow. Our method transports points to a target manifold rather than classifying edges, accelerating inference by orders of magnitude.
• Figure 2: Visualizing the Deterministic Linear Flow. The panels illustrate the evolution of point sets from the initial random configuration $x_0$ ($t=0$) to the target solution manifold $x_1$ ($t=1$). The overlaid vector field indicates the flow direction $x_1 - x_0$, guiding the nodes into a structured circle. As shown in the final panel, the optimal node permutation is recovered by sorting the transported points based on their angular position $\theta$ relative to the origin.
• Figure 3: Geometric Coupling. We construct a specific target $Y$ for each input $X$ based on the optimal tour, creating a deterministic, data-dependent coupling $(x_0, x_1)$.
• Figure 4: Inference Latency vs. Optimality Gap (TSP-50 to TSP-1000). We plot the time--accuracy Pareto frontier on a log--log scale (lower-left is better); marker sizes indicate problem size $N$. CycFlow (red stars) occupies a previously unreachable sub-second regime, achieving competitive optimality gaps while reducing inference latency by orders of magnitude compared to diffusion-based solvers (e.g., DIFUSCO), constructive methods (POMO), and exact solvers (Concorde). This demonstrates an expansion of the efficiency--accuracy Pareto frontier rather than a simple trade-off between speed and solution quality.