Tensor-Network Formulation of the Traveling Salesman Problem and Variants

TL;DR

This paper introduces a tensor-network approach to model and solve the Traveling Salesman Problem and its variants, providing a new formalism that combines exact and heuristic methods.

Contribution

It develops a tensor-network formulation for TSP, enabling explicit marginal formulas and adaptable heuristics for various problem variants.

Findings

The tensor-network method can identify optimal tours in small instances.

The approach is adaptable to several TSP variants and industrial problems.

Experiments compare the method against classical solvers, illustrating its potential and limitations.

Abstract

This work presents a tensor-network formulation of the Traveling Salesman Problem (TSP) and several of its variants. The approach represents candidate tours with tensor-network layers, weights them by Boltzmann factors, and enforces constraints through explicit counting filters. This formalism also yields an explicit tensor-network marginal formula whose zero-temperature, exact-arithmetic limit identifies an optimal feasible tour through a sequential marginal rule. At finite $\tau$ and finite precision, the implemented extraction is a heuristic whose behavior depends on numerical contrast, calibration, and near-degeneracies. We adapt the construction to several generalizations of the TSP and apply it to the Job Reassignment Problem, as a representative industrial integration. The experiments are deliberately small and illustrative; they contextualize the method against exact and heuristic references but do not establish general computational superiority over specialized classical solvers.