Constructing self-referential instances for the clique problem

TL;DR

This paper investigates the inherent difficulty of the clique problem by constructing self-referential graph instances at a phase transition point, revealing why exhaustive search is often necessary.

Contribution

It introduces a novel method of constructing self-referential graph instances at the phase transition, explaining the algorithmic hardness of the clique problem.

Findings

Identifies a phase transition in Erdős–Rényi graphs for the clique problem.

Constructs graph families with identical parameters but different clique properties.

Explains the necessity of exhaustive search due to symmetric transformations.

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.