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~\cite{schuetz2022combinatorial},\cite{schuetz2022graph}. The publication~\cite{schuetz2022combinatorial} stirred a debate whether GNN based method was adequately benchmarked against best prior methods. In particular, critical commentaries~\cite{angelini2023modern} and~\cite{boettcher2023inability} 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~\cite{schuetz2023reply} 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~\cite{schuetz2022combinatorial},\cite{angelini2023modern},\cite{boettcher2023inability},\cite{schuetz2023reply}. 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 \emph{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~\cite{angelini2023modern} and~\cite{boettcher2023inability}.

翻译：近期，基于图神经网络（GNN）的算法被提出用于解决多种组合优化问题，包括最大割问题、最大独立集问题及其他类似问题~\cite{schuetz2022combinatorial},\cite{schuetz2022graph}。文献~\cite{schuetz2022combinatorial}引发了关于基于GNN的方法是否与最佳已有方法进行了充分基准测试的争论。特别是，评论性文章~\cite{angelini2023modern}和~\cite{boettcher2023inability}指出，在随机图设定中，简单的贪心算法性能优于GNN，而采用更复杂的方法实际上可以达到更强的算法性能。作者在回应~\cite{schuetz2023reply}中指出，通过进一步优化参数，GNN的性能可以得到提升。我们无意讨论文献~\cite{schuetz2022combinatorial}、\cite{angelini2023modern}、\cite{boettcher2023inability}、\cite{schuetz2023reply}中论点与反论点的优劣。相反，本文旨在为这些参考文献所考虑的随机图上运行的GNN建立一项基本限制，该限制适用于广泛的GNN架构选择。这些限制源于重叠间隙性质（OGP）相变的存在，这是许多算法（包括经典算法和量子算法）面临的障碍。正如我们在本文中所论证的，由于GNN的局部结构，它同样成为其障碍。我们注意到，与此同时，从简单贪心算法到基于消息传递的更复杂算法，这些已知算法在OGP相变之前能为这些问题提供最佳结果。这使得GNN几乎无法超越已知算法，基于此，我们赞同文献~\cite{angelini2023modern}和~\cite{boettcher2023inability}中得出的结论。