A Fast Heuristic Algorithm Based on Verification and Elimination Methods for Maximum Clique Problem

TL;DR

This paper introduces a fast heuristic algorithm for the maximum clique problem that uses verification and elimination methods to efficiently identify the largest clique in a graph, significantly reducing computation time.

Contribution

The paper presents a novel heuristic algorithm based on verification and elimination techniques that operates in polynomial time for solving the maximum clique problem.

Findings

Successfully applied to random graphs and DIMACS benchmark graphs

Runs in polynomial time, significantly faster than exhaustive methods

Effectively eliminates non-clique subgraphs to focus on promising candidates

Abstract

A clique in an undirected graph G= (V, E) is a subset V' V of vertices, each pair of which is connected by an edge in E. The clique problem is an optimization problem of finding a clique of maximum size in graph. The clique problem is NP-Complete. We have succeeded in developing a fast algorithm for maximum clique problem by employing the method of verification and elimination. For a graph of size N there are 2N sub graphs, which may be cliques and hence verifying all of them, will take a long time. Idea is to eliminate a major number of sub graphs, which cannot be cliques and verifying only the remaining sub graphs. This heuristic algorithm runs in polynomial time and executes successfully for several examples when applied to random graphs and DIMACS benchmark graphs.