On Size and Hardness Generalization in Unsupervised Learning for the Travelling Salesman Problem

TL;DR

The results show that training with larger instance sizes and increasing embedding dimensions can build a more effective representation, enhancing the model's ability to solve TSP.

Abstract

We study the generalization capability of Unsupervised Learning in solving the Travelling Salesman Problem (TSP). We use a Graph Neural Network (GNN) trained with a surrogate loss function to generate an embedding for each node. We use these embeddings to construct a heat map that indicates the likelihood of each edge being part of the optimal route. We then apply local search to generate our final predictions. Our investigation explores how different training instance sizes, embedding dimensions, and distributions influence the outcomes of Unsupervised Learning methods. Our results show that training with larger instance sizes and increasing embedding dimensions can build a more effective representation, enhancing the model's ability to solve TSP. Furthermore, in evaluating generalization across different distributions, we first determine the hardness of various distributions and explore how different hardnesses affect the final results. Our findings suggest that models trained on harder instances exhibit better generalization capabilities, highlighting the importance of selecting appropriate training instances in solving TSP using Unsupervised Learning.

Figures (11)

• Figure 1: Illustration of $\mathbb{T}$ and the corresponding $\mathcal{H}$. $p_1$[1] = $p_2$[3] = $p_3$[2] = $p_4$[5] = $p_5$[4] = 1. This means there is a corresponding Hamiltonian Cycle: $1\rightarrow 3 \rightarrow 2 \rightarrow 5 \rightarrow 4 \rightarrow 1$.
• Figure 2: Training history of different $m$ (embedding dimension) on TSP-2000.
• Figure 3: Training history of different $n$ (instance size) with same embedding dimension $m=1500$.
• Figure 4: Expansion
• Figure 5: Explosion