In this paper, we propose constructing self-referential instances to reveal the inherent algorithmic hardness of the clique problem. First, we prove the existence of a phase transition phenomenon for the clique problem in the Erdős--Rényi random graph model and derive an exact location for the transition point. Subsequently, at the transition point, we construct a family of graphs. In this family, each graph shares the same number of vertices, number of edges, and degree sequence, yet both instances containing a $k$-clique and instances without any $k$-clique are included. These two states can be transformed into each other through a symmetric transformation that preserves the degree of every vertex. This property explains why exhaustive search is required in the critical region: an algorithm must search nearly the entire solution space to determine the existence of a solution; otherwise, a counterinstance can be constructed from the original instance using the symmetric transformation. Finally, this paper elaborates on the intrinsic reason for this phenomenon from the independence of the solution space.

翻译：本文提出通过构建自指实例来揭示团问题固有的算法难度。首先，我们证明了Erdős--Rényi随机图模型中团问题存在相变现象，并推导出相变点的精确位置。随后，在相变点处我们构造了一个图族。该图族中每个图具有相同的顶点数、边数及度序列，但既包含存在$k$-团的实例，也包含不存在任何$k$-团的实例。这两种状态可通过保持每个顶点度数的对称变换相互转换。该性质解释了为何在临界区域需要穷举搜索：算法必须搜索几乎整个解空间才能判定解的存在性；否则，可通过对称变换从原实例构造出反例。最后，本文从解空间的独立性角度阐述了该现象的内在成因。