Constructing self-referential instances for the clique problem
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.