Learning the travelling salesperson problem requires rethinking generalization

Abstract

End-to-end training of neural network solvers for graph combinatorial optimization problems such as the Travelling Salesperson Problem (TSP) have seen a surge of interest recently, but remain intractable and inefficient beyond graphs with few hundreds of nodes. While state-of-the-art learning-driven approaches for TSP perform closely to classical solvers when trained on trivially small sizes, they are unable to generalize the learnt policy to larger instances at practical scales. This work presents an end-to-end neural combinatorial optimization pipeline that unifies several recent papers in order to identify the inductive biases, model architectures and learning algorithms that promote generalization to instances larger than those seen in training. Our controlled experiments provide the first principled investigation into such zero-shot generalization, revealing that extrapolating beyond training data requires rethinking the neural combinatorial optimization pipeline, from network layers and learning paradigms to evaluation protocols. Additionally, we analyze recent advances in deep learning for routing problems through the lens of our pipeline and provide new directions to stimulate future research.

Deep Policy Dynamic Programming for Vehicle Routing Problems

“…After the first DNN model was trained (using example solutions) to construct TSP tours [64], many improvements have been proposed, e.g. different training strategies such as reinforcement learning (RL) [6,14,33,37] and model architectures, which enabled the same idea to be used for other routing problems [15,18,36,45,50,54,67]. Most constructive neural methods are auto-regressive, evaluating the model many times to predict one node at the time, but other works have considered predicting a heatmap of promising edges at once [19,32,52], which allows a tour to be constructed (using sampling or beam search) without further evaluating the model.…”

Learning to Solve Vehicle Routing Problems: A Survey

“…Both methods utilize advanced deep learning models such as Graph Neural Networks (GNNs) [36] and Graph Convolution Networks [37] to extract features of a graph and deploy Memory Augmented Neural Networks [38] and Recurrent Neural Networks (RNNs) [39] to pass sequential information. Both methods require training separate sets of parameters for different graph sizes to produce near-optimal solutions for TSP [40,41]. Table I summarizes studies focused on end-to-end supervised learning.…”

“…Independently, [48] proposed a novel graph representation called Structure2Vec that can encode both the graph and the partial solution at any time step. [44] proposes fully attention-based encoder introducing transformers [49] to solve VRPs, while [41] uses GNNs, a deep learning model dedicated to learn graph information. In return, other studies adopt the proposed encoders including Pointer Networks [50], multi-head attention [51,52,53,54], recurrent neural networks (RNNs) [55], Structure2vec [56,57] and others.…”

“…[96] proposes to learn heatmaps for TSP on small sizes and apply them to arbitrarily large instances. Also, [41] investigates the generalization capabilities of supervised and reinforcement learning methods in end-to-end settings on TSP. The paper has shown that reinforcement learning is better than supervised learning because it does not learn from an existing solution.…”

Intelligent job shop scheduling via deep reinforcement learning over graphs

“…In most existing research for VRP, fully connected undirected graphs are typical for modeling the mutual relationship between customers and vehicles. Such graphical representation facilitates a series of algorithms to exploit the graph neural networks (GNN) to learn the problem representations for solving VRP [9,81,102]. However, this dense topological structure is not applicable for JSSP since it can not describe the precedent constraints among operations.…”