[Paper Review] A Novel Approach Applied to the Largest Clique Problem

Vladimir Gudkov, S. Nussinov+1 more|arXiv (Cornell University)|Sep 17, 2002

Advanced Graph Theory Research3 citations

TL;DR

This paper introduces a novel dynamical approach using high-dimensional physical point systems to solve the NP-complete largest clique problem. By embedding graph vertices into an N−1 dimensional simplex and evolving them under tuned attractive/repulsive forces based on graph connectivity, the method efficiently identifies perfect cliques through energy minimization, achieving near-perfect reconstruction even in complex, overlapping graph structures with high background connectivity.

ABSTRACT

A novel approach to complex problems has been previously applied to graph classification and the graph equivalence problem. Here we apply it to the NP complete problem of finding the largest perfect clique within a graph $G$.

Motivation & Objective

• To address the NP-complete problem of finding the largest perfect clique in a graph using a novel physical dynamics-based method.
• To overcome limitations of traditional combinatorial search by leveraging high-dimensional embedding to avoid local minima.
• To demonstrate that a single simplex distortion algorithm (SDA) can reconstruct perfect cliques even under adverse conditions like high background connectivity and overlapping cliques.
• To validate the method’s robustness across diverse graph topologies, including those with competing cliques and dense random backgrounds.

Proposed method

• Vertices of the graph are mapped to points in an (N−1)-dimensional simplex, initialized at equal distances to ensure symmetry.
• Attractive forces are applied between connected vertices, and repulsive forces between disconnected pairs, with relative strength tuned to v/n, where v is the average vertex degree.
• The system evolves under first-order Aristotelian dynamics, allowing points to move in response to net forces, with energy minimization favoring dense clusters.
• The distance matrix R_ij(t) between points at time t encodes topological relationships, with close points indicating high connectivity.
• To prevent unbounded motion, initial constraints on point radii were relaxed in later runs to avoid artificial local minima from normal reaction forces.
• The method reconstructs cliques by observing convergence of points in the dynamical system, with performance tuned via the U_rep/U_att ratio.

Experimental results

Research questions

• RQ1 Can a high-dimensional physical model based on force dynamics reliably identify the largest perfect clique in a graph?
• RQ2 How does the system perform when multiple overlapping cliques and high background connectivity are present?
• RQ3 What is the impact of force ratio tuning (U_rep/U_att) on clique reconstruction accuracy and convergence?
• RQ4 Does the absence of radial constraints in the dynamical system eliminate spurious local minima that hinder convergence?
• RQ5 Can the method reconstruct the largest clique even when some vertices are missing one connection to the clique?

Key findings

• The method successfully reconstructed a 100-vertex perfect clique in a 400-vertex graph with 70% background connectivity, even with overlapping 80-vertex cliques.
• Doubling the repulsive force ratio (U_rep/U_att) from 1 to 2 significantly improved reconstruction fidelity, as shown in Figure 13.
• The system achieved full reconstruction of the maximal clique regardless of background connectivity or the presence of partially overlapping smaller cliques.
• Relaxing the radial constraint on point positions eliminated artificial local minima, enabling convergence to the true global minimum energy state.
• Theoretical analysis confirms that in N−1 dimensions, the net force on any test point never vanishes, preventing false local minima under ideal conditions.
• The method remains effective even in the worst-case scenario where a vertex is missing only one connection to the clique, provided U_rep > (n−1)·U_att.