A GREAT Architecture for Edge-Based Graph Problems Like TSP

In the last years, many neural network-based approaches have been proposed to tackle combinatorial optimization problems such as routing problems. Many of these approaches are based on graph neural networks (GNNs) or related transformers, operating on the Euclidean coordinates representing the routing problems. However, GNNs are inherently not well suited to operate on dense graphs, such as in routing problems. Furthermore, models operating on Euclidean coordinates cannot be applied to non-Euclidean versions of routing problems that are often found in real-world settings. To overcome these limitations, we propose a novel GNN-related edge-based neural model called Graph Edge Attention Network (GREAT). We evaluate the performance of GREAT in the edge-classification task to predict optimal edges in the Traveling Salesman Problem (TSP). We can use such a trained GREAT model to produce sparse TSP graph instances, keeping only the edges GREAT finds promising. Compared to other, non-learning-based methods to sparsify TSP graphs, GREAT can produce very sparse graphs while keeping most of the optimal edges. Furthermore, we build a reinforcement learning-based GREAT framework which we apply to Euclidean and non-Euclidean asymmetric TSP. This framework achieves state-of-the-art results.

Less Is More -- On the Importance of Sparsification for Transformers and Graph Neural Networks for TSP

Most of the recent studies tackling routing problems like the Traveling Salesman Problem (TSP) with machine learning use a transformer or Graph Neural Network (GNN) based encoder architecture. However, many of them apply these encoders naively by allowing them to aggregate information over the whole TSP instances. We, on the other hand, propose a data preprocessing method that allows the encoders to focus on the most relevant parts of the TSP instances only. In particular, we propose graph sparsification for TSP graph representations passed to GNNs and attention masking for TSP instances passed to transformers where the masks correspond to the adjacency matrices of the sparse TSP graph representations. Furthermore, we propose ensembles of different sparsification levels allowing models to focus on the most promising parts while also allowing information flow between all nodes of a TSP instance. In the experimental studies, we show that for GNNs appropriate sparsification and ensembles of different sparsification levels lead to substantial performance increases of the overall architecture. We also design a new, state-of-the-art transformer encoder with ensembles of attention masking. These transformers increase model performance from a gap of $0.16\%$ to $0.10\%$ for TSP instances of size 100 and from $0.02\%$ to $0.00\%$ for TSP instances of size 50.