[Paper Review] An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem

Chaitanya K. Joshi, Thomas Laurent+1 more|arXiv (Cornell University)|Jun 4, 2019

Vehicle Routing Optimization MethodsEngineering43 references202 citations

TL;DR

The paper proposes a non-autoregressive graph ConvNet model that outputs a TSP tour heat-map and uses beam search to obtain valid tours, achieving higher solution quality and faster inference than autoregressive deep learning methods on fixed-size 2D Euclidean TSP 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.

Motivation & Objective

• Motivate learning-based solutions for the NP-hard Travelling Salesman Problem (TSP) on 2D Euclidean graphs.
• Develop a graph convolutional network that directly outputs a tour-adjacency heat-map.
• Enable fast, parallelizable, non-autoregressive inference via beam search.
• Train the model supervisedly using optimal solutions from Concorde to improve sample efficiency.
• Compare against autoregressive DL methods and traditional OR solvers to assess quality and speed.

Proposed method

• Build a graph ConvNet that processes node coordinates and edge distances to produce per-edge features.
• Predict an edge adjacency heat-map via an MLP applied to edge embeddings.
• Train end-to-end with cross-entropy loss using ground-truth TSP tours from Concorde.
• Convert the predicted heat-map into a valid tour with post-hoc beam search (and variants with shortest-tour heuristic).
• Use k-NN graph embeddings and residual graph conv layers to capture graph structure and anisotropic diffusion.

Experimental results

Research questions

• RQ1 Can a non-autoregressive graph ConvNet directly predict TSP tour edges effectively for 2D Euclidean graphs?
• RQ2 Does beam search decoding on a heat-map adjacency representation yield competitive tours compared to autoregressive models?
• RQ3 How do solution quality, inference speed, and sample efficiency compare to existing DL approaches and traditional OR solvers?
• RQ4 What is the model’s generalization behavior across fixed graph sizes and to other sizes?

Key findings

• The method reduces the average optimality gap from 0.52% to 0.01% for 50 nodes.
• The method reduces the average optimality gap from 2.26% to 1.39% for 100 nodes.
• Inference is fast due to GPU-accelerated, highly parallelized graph ConvNet and beam search.
• Supervised training with optimal solutions is more sample-efficient than reinforcement learning in this setup.
• Beam search with 1,280 solutions outperforms autoregressive deep learning methods in both quality and speed on fixed-size graphs.