Barriers for the performance of graph neural networks (GNN) in discrete random structures

Recently, graph neural network (GNN)-based algorithms were proposed to solve a variety of combinatorial optimization problems [M. J. Schuetz, J. K. Brubaker, H. G. Katzgraber, Nat. Mach. Intell.4, 367-377 (2022)]. GNN was tested in particular on randomly generated instances of these problems. The publication [M. J. Schuetz, J. K. Brubaker, H. G. Katzgraber, Nat. Mach. Intell.4, 367-377 (2022)] stirred a debate whether the GNN-based method was adequately benchmarked against best prior methods. In particular, critical commentaries [M. C. Angelini, F. Ricci-Tersenghi, Nat. Mach. Intell.5, 29-31 (2023)] and [S. Boettcher, Nat. Mach. Intell.5, 24-25 (2023)] point out that a simple greedy algorithm performs better than the GNN. We do not intend to discuss the merits of arguments and counterarguments in these papers. Rather, in this note, we establish a fundamental limitation for running GNN on random instances considered in these references, for a broad range of choices of GNN architecture. Specifically, these barriers hold when the depth of GNN does not scale with graph size (we note that depth 2 was used in experiments in [M. J. Schuetz, J. K. Brubaker, H. G. Katzgraber, Nat. Mach. Intell.4, 367-377 (2022)]), and importantly, these barriers hold regardless of any other parameters of GNN architecture. These limitations arise from the presence of the overlap gap property (OGP) phase transition, which is a barrier for many algorithms, including importantly local algorithms, of which GNN is an example. At the same time, some algorithms known prior to the introduction of GNN 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 [M. C. Angelini, F. Ricci-Tersenghi, Nat. Mach. Intell.5, 29-31 (2023)] and [S. Boettcher, Nat. Mach. Intell.5, 24-25 (2023)].