Tensor Network Generator-Enhanced Optimization for Traveling Salesman Problem

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.

Figures (7)

• Figure 1: Schematic picture of GEO for TSP. The framework iteratively refines a generative model by training on a softmax-weighted distribution of candidate solutions. At each iteration, the model generates new samples that are biased toward lower-cost tours.
• Figure 2: Effect of bond dimension on sampling quality for ulysses16. Tour length distributions are shown for the initial random population, the softmax-weighted target distribution, and MPS samples at bond dimensions $\in \{4, 16, 64, 128\}$. Higher bond dimensions enable the MPS to better approximate the target distribution. All distributions consist of $2^{16}$ samples.
• Figure 3: Evolution of tour length distribution across GEO iterations for ulysses16. The distribution progressively shifts toward shorter tour lengths as iterations proceed.
• Figure 4: Convergence of best tour length for ulysses16 (left, 16 cities) and att48 (right, 48 cities). TN-GEO converges to or near the optimal solution, outperforming both swap and 2-opt heuristics.
• Figure 5: Best tours found for ulysses16 (left three) and att48 (right three). For comparison, the tours that are found by the swap and 2-opt heuristics are also shown.