Abstract
The Maximum Clique Problem (MCP) is a classical NP-hard problem that has gained considerable attention due to its numerous real-world applications and theoretical complexity. It is inherently computationally complex, and so exact methods may require prohibitive computing time. Nature-inspired meta-heuristics have proven their utility in solving many NP-hard problems. In this research, we propose a simulated annealing-based algorithm that we call Clique Finder algorithm to solve the MCP. Our algorithm uses a logarithmic cooling schedule and two moves that are selected in an adaptive manner. The objective (error) function is the total number of missing links in the clique, which is to be minimized. The proposed algorithm was evaluated using benchmark graphs from the open-source library DIMACS, and results show that the proposed algorithm had a high success rate.
Highlights
The Maximum Clique Problem (MCP) is a classical NP-hard combinatorial optimization problem concerned with finding the largest subset in a graph, where all nodes in the subset share an edge.Given an undirected graph G (V,E ), a clique C is a subset of the graph, such that there is an edge between any two vertices in C
We studied nature-inspired methods that were applied to MCP such as the harmony search algorithm, ant colony optimization algorithm, genetic algorithm, intelligent water drop algorithm, and simulated annealing algorithm
We found that the harmony search and intelligent water drops algorithms require many parameters
Summary
The Maximum Clique Problem (MCP) is a classical NP-hard combinatorial optimization problem concerned with finding the largest subset in a graph, where all nodes in the subset share an edge. Given an undirected graph G (V,E ) , a clique C is a subset of the graph, such that there is an edge between any two vertices in C. A clique C is said to be maximal if it is not a subset of any larger clique of G. The clique in G with the largest cardinality is known as a maximum clique, i.e. it cannot be extended to a larger one. The MCP seeks a maximum clique in a graph. A related problem is the maximum weighted clique problem: an NP-complete problem to find the clique with the maximum weight sum in a given undirected graph
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