An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem

TL;DR

This work tackles the 2D Euclidean Travelling Salesman Problem by introducing a non-autoregressive Graph ConvNet that outputs a heat-map of edge probabilities and uses beam search to form tours. Trained in a supervised manner on Concorde-provided optimal tours, the method achieves superior solution quality and faster inference than autoregressive DL approaches on fixed-size graphs (20, 50, 100 nodes). While competitive with other learning-based methods, it remains outperformed by exact OR solvers, and its generalization to varying problem sizes is limited. The approach highlights strong potential for parallelizable graph-based solvers and points to future work in transfer learning and reinforcement learning to tackle larger instances.

Abstract

This paper introduces a new learning-based approach for approximately solving the Travelling Salesman Problem on 2D Euclidean graphs. We use deep Graph Convolutional Networks to build efficient TSP graph representations and output tours in a non-autoregressive manner via highly parallelized beam search. Our approach outperforms all recently proposed autoregressive deep learning techniques in terms of solution quality, inference speed and sample efficiency for problem instances of fixed graph sizes. In particular, we reduce the average optimality gap from 0.52% to 0.01% for 50 nodes, and from 2.26% to 1.39% for 100 nodes. Finally, despite improving upon other learning-based approaches for TSP, our approach falls short of standard Operations Research solvers.

Figures (8)

• Figure 1: Overview of our approach. Taking a 2D graph as input, the graph ConvNet model outputs an edge adjacency matrix denoting the probabilities of edges occurring on the TSP tour. This is converted to a valid tour using beam search. All components are highly parallelized and solutions are produced in a one-shot, non-autoregressive manner.
• Figure 2: Validation optimality gap vs. Number of training samples for our approach (using beam search with beam width 1,280) and kool2018attention (sampling 1,280 solutions).
• Figure 3: The proposed graph convolution layer for computing $h$-dimensional representations $x_i$ for each node $i$ and $e_{ij}$ for the edge between each node $i$ and $j$ in the graph. The red and blue arrows respectively represent the node and edge information used to compute their representation at the next layer. Multiple layers of graph convolution are applied to progressively extract more and more compositional features of the input graph.
• Figure 4: Learning rate or Training/Validation loss vs. Number of training samples.
• Figure 5: Impact of model architecture on validation optimality gap.