A heuristic based harmony search algorithm for maximum clique problem

Abstract

The maximum clique problem (MCP) is to determine a complete subgraph (clique) of maximum cardinality in a given graph. MCP is conspicuous for having real world applications and for its potentiality of modeling other combinatorial problems and is one of the most studied NP-hard problems. This paper investigates the capabilities of Harmony Search (HS) algorithm, a music inspired meta heuristic for solving maximum clique problem. We propose and compare two different instantiations of a generic HS algorithm namely Harmony Search for MCP (HS_MCP) and Harmony Search with idiosyncratic harmonies for MCP (HSI_MCP) for this problem. HS_MCP has better exploitation and inferior exploration capabilities than HSI_MCP whereas HSI_MCP has better exploration and inferior exploitation capabilities than HSI_MCP, it has been concluded that former performs better than latter by testing them on all the instances of DIMACS benchmark graphs. HS_MCP has been compared with a recently proposed Harmony search based algorithm for MCP called Binary Harmony search (BHS) and the simulation results show that HS_MCP significantly outperforms BHS in terms of solution quality. The asymptotic time complexity of HS_MCP is \(O(G \times N^3)\) where G is the number of generations and N is the number of nodes in the graph. A glimpse of effectiveness of some state-of-the-art exact algorithms on MCP has also been provided.