Tensor Network Generator-Enhanced Optimization for Traveling Salesman Problem
arXiv:2602.20175v1 Announce Type: new 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. Experiment
arXiv:2602.20175v1 Announce Type: new 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.
Executive Summary
This article presents an innovative application of the tensor network generator-enhanced optimization (TN-GEO) framework to the traveling salesman problem (TSP). The authors employ a tensor network Born machine based on automatically differentiable matrix product states (MPS) as the generative model. This approach offers significant advantages over traditional binary encoding methods, including reduced variable requirements and guaranteed valid tour construction. Experimental results demonstrate that TN-GEO outperforms classical heuristics on TSPLIB benchmark instances. The article also introduces a $k$-site MPS variant that learns distributions over $k$-grams, enabling parameter-efficient modeling for larger instances. The authors' findings have significant implications for the field of combinatorial optimization and may lead to improved solution methods for TSP and related problems.
Key Points
- ▸ TN-GEO is applied to the TSP using a tensor network Born machine based on MPS
- ▸ The approach employs permutation-based formulation with integer variables and autoregressive sampling with masking
- ▸ The $k$-site MPS variant learns distributions over $k$-grams and enables parameter-efficient modeling
Merits
Strength in Addressing TSP
TN-GEO effectively addresses the TSP by leveraging the power of tensor networks and autoregressive sampling, leading to improved solution quality and efficiency
Parameter-Efficient Modeling
The $k$-site MPS variant enables parameter-efficient modeling for larger instances, making it a promising approach for real-world applications
Guaranteed Valid Tour Construction
The use of autoregressive sampling with masking ensures that every generated sample is a valid tour, eliminating the need for penalty terms
Demerits
Limited Scalability
The approach may face scalability challenges for extremely large TSP instances, requiring further research on scalable tensor network architectures
Dependence on High-Performance Computing
TN-GEO's computational complexity may be high, requiring access to high-performance computing resources for large-scale TSP instances
Expert Commentary
The authors' innovative application of TN-GEO to the TSP is a significant contribution to the field of combinatorial optimization. The approach's strengths, including parameter-efficient modeling and guaranteed valid tour construction, make it a promising candidate for solving large-scale TSP instances. However, the limitations, particularly scalability and dependence on high-performance computing, require further research to fully realize the potential of TN-GEO. As the field continues to evolve, it is essential to explore the applicability of TN-GEO to other combinatorial optimization problems and investigate methods to enhance its scalability.
Recommendations
- ✓ Further research is needed to develop scalable tensor network architectures for large-scale TSP instances
- ✓ The authors should investigate methods to reduce the computational complexity of TN-GEO and make it more accessible to researchers without high-performance computing resources
