TSP with Predictions: Heatmap to Tour with Provable Guarantees

2026-07-04

AI Summary

This work addresses the challenge of efficiently and reliably converting neural network–generated edge existence probability heatmaps into valid tours for the Traveling Salesman Problem (TSP). To this end, the authors propose a novel algorithm that integrates machine learning predictions with classical combinatorial optimization within the "algorithms with predictions" framework. Notably, this is the first approach for heatmap-based TSP solvers to offer theoretical approximation guarantees: the solution quality achieves a $(1 + 2\eta/\text{OPT})$-approximation, where $\eta$ denotes the L1 distance between the predicted heatmap and the optimal tour. Experimental results demonstrate that the method outperforms existing conversion strategies lacking theoretical guarantees while maintaining computational efficiency, and further confirm the tightness of the derived theoretical bound.

Abstract

The Traveling Salesperson Problem (TSP) has long served as a benchmark for evaluating the strength of optimization techniques in the classical theory of algorithms. In recent efforts to apply ML to algorithmic problems, TSP has also become a natural testbed for the development of ML-based techniques. A common approach is to train a neural network to output a heatmap estimating the likelihood of each edge to be part of the optimal tour; however, converting such a heatmap into an actual tour remains a non-trivial and often computationally intensive step. In this work, we propose algorithms for transforming heatmaps into tours with theoretical guarantees linking the achieved approximation ratio to the quality of the provided heatmap. In the spirit of algorithms with predictions, our results can be described as $(1+2\fracη{\mathrm{OPT}})$-approximation algorithms, where $η$ denotes the L1 distance between the prediction (heatmap) and an optimal solution (tour). Since the previous works lack such explicit guarantees, we compare our approach against them experimentally.

Problem

• Traveling Salesperson Problem
• heatmap
• algorithmic guarantees
• approximation ratio
• predictions

Innovation

• TSP with predictions
• heatmap-to-tour conversion
• approximation guarantees
• algorithms with predictions
• L1 error analysis