Tensor Network Generator-Enhanced Optimization for Traveling Salesman Problem

TL;DR

This paper introduces a tensor network-based generative optimization framework for solving the traveling salesman problem, leveraging permutation-based models and autoregressive sampling to efficiently generate valid tours and outperform classical heuristics.

Contribution

The paper proposes a novel tensor network generator-enhanced optimization method using permutation-based models and autoregressive sampling for TSP, with a $k$-site MPS variant for larger instances.

Findings

Outperforms classical heuristics on TSPLIB instances with up to 52 cities.

The $k$-site MPS variant improves results by focusing on local correlations.

Demonstrates the effectiveness of tensor network models in combinatorial optimization.

Abstract

We present an application of the tensor network generator-enhanced optimization (TN-GEO) framework to address the traveling salesman problem (TSP), a fundamental combinatorial optimization challenge. Our approach employs a tensor network Born machine based on automatically differentiable matrix product states (MPS) as the generative model, using the Born rule to define probability distributions over candidate solutions. Unlike approaches based on binary encoding, which require $N^2$ variables and penalty terms to enforce valid tour constraints, we adopt a permutation-based formulation with integer variables and use autoregressive sampling with masking to guarantee that every generated sample is a valid tour by construction. We also introduce a $k$-site MPS variant that learns distributions over $k$-grams (consecutive city subsequences) using a sliding window approach, enabling parameter-efficient modeling for larger instances. Experimental validation on TSPLIB benchmark instances with up to 52 cities demonstrates that TN-GEO can outperform classical heuristics including swap and 2-opt hill-climbing. The $k$-site variants, which put more focus on local correlations, show better results compared to the full-MPS case.