Inability of a graph neural network heuristic to outperform greedy algorithms in solving combinatorial optimization problems like Max-Cut

TL;DR

This paper critically evaluates a graph neural network heuristic for NP-hard problems like Max-Cut, showing it does not outperform simple greedy algorithms and highlighting evaluation misconceptions.

Contribution

It provides a rigorous analysis demonstrating the limitations of GNN heuristics in combinatorial optimization, challenging prior claims of their effectiveness.

Findings

GNN heuristic performs only slightly better than gradient descent.

Greedy algorithms outperform the GNN heuristic in Max-Cut.

Highlights misconceptions in heuristic evaluation methods.

Abstract

In Nature Machine Intelligence 4, 367 (2022), Schuetz et al provide a scheme to employ graph neural networks (GNN) as a heuristic to solve a variety of classical, NP-hard combinatorial optimization problems. It describes how the network is trained on sample instances and the resulting GNN heuristic is evaluated applying widely used techniques to determine its ability to succeed. Clearly, the idea of harnessing the powerful abilities of such networks to ``learn'' the intricacies of complex, multimodal energy landscapes in such a hands-off approach seems enticing. And based on the observed performance, the heuristic promises to be highly scalable, with a computational cost linear in the input size $n$, although there is likely a significant overhead in the pre-factor due to the GNN itself. However, closer inspection shows that the reported results for this GNN are only minutely better than those for gradient descent and get outperformed by a greedy algorithm, for example, for Max-Cut. The discussion also highlights what I believe are some common misconceptions in the evaluations of heuristics.