Constructing self-referential instances for the clique problem

TL;DR

This work addresses the inherent hardness of the clique problem by proving a precise phase transition for the existence of a $k$-clique in $G(n,m)$ and constructing self-referential, degree-preserving graph instances at the transition. The authors combine first- and second-moment analyses to locate the threshold at $m = rac{n(n-1)}{2} obreak\cdot n^{-rac{2}{k-1}}$, showing whp nonexistence below and whp existence above it. They then build a class of graphs sharing $|V|$, $|E|$, and degree sequence that can be transformed between having and lacking a $k$-clique via symmetric operations, with the transformed instance unlikely to retain a clique, thereby forcing exhaustive search in the critical regime. The results illuminate why solution-space independence necessitates exhaustive verification at the phase transition and suggest that this self-referential construction approach can extend to other combinatorial problems.

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.

Figures (3)

• Figure 1: In Figure (a), the solid lines represent the original edges in the graph, and each vertex has a degree of 1. After the transformation, the new graph is shown in Figure (b). Following the establishment of these new edges, each vertex still maintains a degree of 1, thereby ensuring degree invariance.
• Figure 2: In figure (a), the blue vertices form a clique of size 5. After the dashed edges are added in figure (b), the clique formed by the blue vertices is disrupted and cannot form a new clique.
• Figure 3: Figure (a) illustrates the current search state of the program. The blue region represents the portion of the graph that has already been searched, while the orange region indicates areas not yet explored. Within the possible set of graphs, two distinct configurations exist: Figure (b) and Figure (c). Figure (b) does not contain a 4-clique, whereas Figure (c) does (formed by the purple vertices). Figures (b) and (c) can be obtained from each other by applying a symmetric transformation within the red region.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition 4.1
• Theorem 4.1