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arXiv · 2601.19393

Constructing self-referential instances for the clique problem

Jiaqi Li·Shuli Hu·Xianxian Li·Minghao Yin
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Abstract

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.

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Jiaqi Li, Shuli Hu, Xianxian Li, Minghao Yin. 2026-02-05. Constructing self-referential instances for the clique problem. https://arxiv.org/abs/2601.19393

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