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}.

本文研究了图神经网络在解决随机图组合优化问题方面的局限性，发现由于“Overlap Gap Property”存在，即使是针对广泛选择的GNN体系结构，也很难超越已有算法的表现。

Barriers for the performance of graph neural networks (GNN) in discrete
  random structures. A comment
  on~\cite{schuetz2022combinatorial},\cite{angelini2023modern},\cite{schuetz2023reply}

离散随机结构下图神经网络(GNN)性能的障碍 - 对文章 《Schuetz等：组合方法的进展》、《Angelini等：现代表面建模》、以及《Schuetz的回复》的评论

离散随机结构下图神经网络(GNN)性能的障碍 - 对文章《Schuetz等：组合方法的进展》、《Angelini等：现代表面建模》、以及《Schuetz的回复》的评论