🤖 AI Summary
This study investigates the intrinsic computational hardness of the clique problem in its critical region. By constructing pairs of graphs at the phase transition point of the Erdős–Rényi random graph model that share identical numbers of vertices, edges, and degree sequences yet exhibit opposite solution statuses—i.e., one contains a k-clique while the other does not—the authors employ degree-preserving symmetric transformations to reveal the indistinguishability of their solution spaces. This work presents the first family of self-referential instances that rigorously establishes the existence of an exact phase transition threshold for the clique problem. Theoretically, it demonstrates that near this threshold, any algorithm must almost surely explore the entire solution space to determine the existence of a solution, thereby explaining the unavoidable necessity of exhaustive search in this regime.
📝 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 Erd\H{o}s--R\'enyi 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.