Graph Neural Networks are Heuristics

TL;DR

This paper shows that graph neural networks can serve as effective, unsupervised heuristics for combinatorial optimization problems like the Traveling Salesman Problem, without the need for search or supervision.

Contribution

It introduces a method where GNNs encode global constraints to generate solutions directly, reframing learning as instantiating heuristics rather than augmenting classical algorithms.

Findings

GNNs can solve TSP without supervision or search.

Dropout and ensembling improve solution diversity and quality.

GNNs internalize global structure to act as heuristics.

Abstract

We demonstrate that a single training trajectory can transform a graph neural network into an unsupervised heuristic for combinatorial optimization. Focusing on the Travelling Salesman Problem, we show that encoding global structural constraints as an inductive bias enables a non-autoregressive model to generate solutions via direct forward passes, without search, supervision, or sequential decision-making. At inference time, dropout and snapshot ensembling allow a single model to act as an implicit ensemble, reducing optimality gaps through increased solution diversity. Our results establish that graph neural networks do not require supervised training nor explicit search to be effective. Instead, they can internalize global combinatorial structure and function as strong, learned heuristics. This reframes the role of learning in combinatorial optimization: from augmenting classical algorithms to directly instantiating new heuristics.