Graph Neural Network Guided Local Search for the Traveling Salesperson Problem

TL;DR

This paper addresses real-time Euclidean TSP by coupling a Graph Neural Network that predicts edge-wise global regret with Guided Local Search (GLS). The authors define global regret, implement a regret approximation model on the line graph $L(G)$ to predict $\hat{r}_{ij}$ for each edge, and integrate these predictions into GLS to guide edge penalization and search. Empirically, the approach converges to high-quality solutions faster than several learning-based baselines, achieving substantial reductions in mean optimality gaps on 20, 50, and 100-node TSP instances and strong generalization to larger and real-world TSPLIB instances. The method emphasizes a simple, edge-focused input (edge weight) and demonstrates that line-graph representation, along with regret-guided perturbations, improves search efficiency and solution quality for real-time routing tasks. The work suggests broad applicability to routing problems beyond TSP through the general notion of global regret and GLS integration.

Abstract

Solutions to the Traveling Salesperson Problem (TSP) have practical applications to processes in transportation, logistics, and automation, yet must be computed with minimal delay to satisfy the real-time nature of the underlying tasks. However, solving large TSP instances quickly without sacrificing solution quality remains challenging for current approximate algorithms. To close this gap, we present a hybrid data-driven approach for solving the TSP based on Graph Neural Networks (GNNs) and Guided Local Search (GLS). Our model predicts the regret of including each edge of the problem graph in the solution; GLS uses these predictions in conjunction with the original problem graph to find solutions. Our experiments demonstrate that this approach converges to optimal solutions at a faster rate than three recent learning based approaches for the TSP. Notably, we reduce the mean optimality gap on the 100-node problem set from 1.534% to 0.705%, a 2x improvement. When generalizing from 20-node instances to the 100-node problem set, we reduce the optimality gap from 18.845% to 2.622%, a 7x improvement.

Figures (6)

• Figure 1: From a TSP formulated as a graph, we take the line graph (a) and input it into our regret approximation model (b), which predicts the regret of including each edge in the solution. GLS (c) uses these predictions in conjunction with the original problem graph to quickly find a high-quality solution.
• Figure 2: An example of a graph and the corresponding line graph. The edges in $G$ are the nodes in $L(G)$, and vice-versa.
• Figure 3: Mean optimality gap and computation time per instance for three increasingly difficult problem sets. The left plot shows non-learning based approaches, where the Nearest Neighbour heuristic, LKH-3, and Concorde typically form the Pareto front. The right plot shows learning based approaches, where attentionlearntosolve and our approach form the Pareto front.
• Figure 4: Solution quality as a function of computation time for three increasingly difficult problem sets, demonstrating that our method converges to optimal solutions at a faster rate than the evaluated learning based approaches. The left plot shows the mean optimality gap. The right plot shows the percentage of optimally solved problems.
• Figure 5: Solution quality as a function of computation time for three increasingly difficult generalization tasks, demonstrating that our method is able to generalize from smaller to larger problem instances better than several learning based approaches. The left plot shows the mean optimality gap. The right plot shows the percentage of optimally solved problems.