An Impatient Evolutionary Algorithm With Probabilistic Tabu Search for Unified Solution of Some NP-Hard Problems in Graph and Set Theory via Clique Finding

Many graph- and set-theoretic problems, because of their tremendous application potential and theoretical appeal, have been well investigated by the researchers in complexity theory and were found to be NP-hard. Since the combinatorial complexity of these problems does not permit exhaustive searches for optimal solutions, only near-optimal solutions can be explored using either various problem-specific heuristic strategies or metaheuristic global-optimization methods, such as simulated annealing, genetic algorithms, etc. In this paper, we propose a unified evolutionary algorithm (EA) to the problems of maximum clique finding, maximum independent set, minimum vertex cover, subgraph and double subgraph isomorphism, set packing, set partitioning, and set cover. In the proposed approach, we first map these problems onto the maximum clique-finding problem (MCP), which is later solved using an evolutionary strategy. The proposed impatient EA with probabilistic tabu search (IEA-PTS) for the MCP integrates the best features of earlier successful approaches with a number of new heuristics that we developed to yield a performance that advances the state of the art in EAs for the exploration of the maximum cliques in a graph. Results of experimentation with the 37 DIMACS benchmark graphs and comparative analyses with six state-of-the-art algorithms, including two from the smaller EA community and four from the larger metaheuristics community, indicate that the IEA-PTS outperforms the EAs with respect to a Pareto-lexicographic ranking criterion and offers competitive performance on some graph instances when individually compared to the other heuristic algorithms. It has also successfully set a new benchmark on one graph instance. On another benchmark suite called Benchmarks with Hidden Optimal Solutions, IEA-PTS ranks second, after a very recent algorithm called COVER, among its peers that have experimented with this suite.