Graph Neural Networks are Heuristics

Yimeng Min; Carla P. Gomes

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arXiv · 2601.13465

Graph Neural Networks are Heuristics

Yimeng Min·Carla P. Gomes

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Abstract

Graph neural networks are usually treated as auxiliaries for combinatorial optimization: they imitate algorithms, guide search, or supply scores to classical procedures. We show that this auxiliary role is not intrinsic. A GNN can itself be a heuristic. For the Euclidean Travelling Salesman Problem, we train a non-autoregressive GNN with no labels, rewards, sequential decoding, search, or local improvement. A differentiable Hamiltonian-cycle objective is the only supervision. The trained model produces a complete tour in one forward pass, while dropout and snapshots from a single training trajectory provide solution diversity without engineered moves. The heuristic is therefore learned, not programmed. It is also fast: batched inference remains in the millisecond regime on GPUs. Experiments on TSP100, TSP200, and TSP500 show that the model consistently improves over nearest-neighbor greedy baselines. These results identify unsupervised GNNs as a class of fast learned heuristics for combinatorial optimization.

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Yimeng Min, Carla P. Gomes. 2026-01-19. Graph Neural Networks are Heuristics. https://arxiv.org/abs/2601.13465

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