Graph Neural Networks are Heuristics
Yimeng Min, Carla P. Gomes
TL;DR
This work shows that graph neural networks can function as learned heuristics for combinatorial optimization without supervision or explicit search. By encoding the Hamiltonian cycle constraint for TSP directly into the learning objective and employing a soft-permutation relaxation, the method solves complete tours in a single forward pass. Key innovations include a symmetry-preserving, equivariant feature extractor, stochasticity via dropout, and snapshot-ensemble inference that yields diverse solutions without extra training. Empirical results demonstrate competitive or superior tour quality with millisecond inference on GPUs, relative to classical heuristics, highlighting a shift toward structure-aware learning as a direct instantiation of new heuristics for combinatorial problems.
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.