Tensor Network Generator-Enhanced Optimization for Traveling Salesman Problem

Researchers Ryo Sakai and Chen-Yu Liu have introduced TN-GEO, a novel AI framework that applies tensor network theory and generative modeling to the classic Traveling Salesman Problem (TSP). This combinatorial optimization challenge, fundamental to logistics and routing, is tackled using a tensor network Born machine based on automatically differentiable Matrix Product States (MPS). Unlike traditional binary-encoding methods that require complex penalty terms, TN-GEO employs a permutation-based formulation with integer variables. It uses autoregressive sampling with masking to guarantee every generated candidate is a valid tour by construction, eliminating the need for post-generation correction.

The technical innovation includes a k-site MPS variant that learns probability distributions over k-grams, or consecutive city subsequences, using a sliding window. This allows for parameter-efficient modeling of local correlations in larger problem instances. Experimental results on standard TSPLIB benchmarks, with instances containing up to 52 cities, demonstrate that the TN-GEO framework can outperform established classical heuristics like swap and 2-opt hill-climbing. The k-site variants, which focus more on local structure, showed better performance than the full-MPS approach. This work bridges techniques from quantum-inspired tensor networks, machine learning, and classical optimization, suggesting a promising new direction for solving complex discrete optimization problems without relying on quantum hardware.