Constructing self-referential instances for the clique problem

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.

💡 Research Summary

The paper introduces a novel construction of “self‑referential instances” for the classic NP‑complete Clique problem, aiming to demonstrate why exhaustive search is unavoidable in the critical region of the problem’s parameter space. The authors begin by establishing the existence of a sharp phase‑transition phenomenon for the presence of a k‑clique in the Erdős–Rényi random graph model G(n,m). Using the first‑moment method, they show that when the edge density ρ is below a certain threshold function W(n), the expected number of k‑cliques tends to zero, implying that almost surely no k‑clique exists. Conversely, they argue that the first‑moment method alone cannot guarantee existence when ρ exceeds the threshold, because a diverging expectation does not preclude a non‑zero probability of absence.

To bridge this gap, the paper applies the second‑moment method. By proving that the variance of the number of k‑cliques is asymptotically negligible compared to the square of its expectation (Var