Barriers for the performance of graph neural networks (GNN) in discrete random structures. A comment on~citeschuetz2022combinatorial,citeangelini2023modern,citeschuetz2023reply

Recently graph neural network (GNN) based algorithms were proposed to solve a variety of combinatorial optimization problems, including Maximum Cut problem, Maximum Independent Set problem and similar other problems~citeschuetz2022combinatorial,citeschuetz2022graph. The publication~citeschuetz2022combinatorial stirred a debate whether GNN based method was adequately benchmarked against best prior methods. In particular, critical commentaries~citeangelini2023modern and~citeboettcher2023inability point out that simple greedy algorithm performs better than GNN in the setting of random graphs, and in fact stronger algorithmic performance can be reached with more sophisticated methods. A response from the authors~citeschuetz2023reply pointed out that GNN performance can be improved further by tuning up the parameters better. We do not intend to discuss the merits of arguments and counter-arguments in~citeschuetz2022combinatorial,citeangelini2023modern,citeboettcher2023inability,citeschuetz2023reply. Rather in this note we establish a fundamental limitation for running GNN on random graphs considered in these references, for a broad range of choices of GNN architecture. These limitations arise from the presence of the Overlap Gap Property (OGP) phase transition, which is a barrier for many algorithms, both classical and quantum. As we demonstrate in this paper, it is also a barrier to GNN due to its local structure. We note that at the same time known algorithms ranging from simple greedy algorithms to more sophisticated algorithms based on message passing, provide best results for these problems up to the OGP phase transition. This leaves very little space for GNN to outperform the known algorithms, and based on this we side with the conclusions made in~citeangelini2023modern and~citeboettcher2023inability.