Finding the Maximal Independent Sets of a Graph Including the Maximum Using a Multivariable Continuous Polynomial Objective Optimization Formulation

We propose a multivariable continuous polynomial optimization formulation to find arbitrary maximal independent sets of any size for any graph. A local optima of the optimization problem yields a maximal independent set, while the global optima yields a maximum independent set. The solution is two phases. The first phase is listing all the maximal cliques of the graph and the second phase is solving the optimization problem. We believe that our algorithm is efficient for sparse graphs, for which there exist fast algorithms to list their maximal cliques. Our algorithm was tested on some of the DIMACS maximum clique benchmarks and produced results efficiently. In some cases our algorithm outperforms other algorithms, such as cliquer

A New Genetic Algorithm for the Maximum Clique Problem

The Maximum Clique Problem is widely studied optimization problem and plays an essential part in computer science. In the literature, therefore, there are many exact and heuristic approaches and new strategies are continuously being proposed. In this paper, the maximum clique problem is studied, and a new hybrid genetic algorithm is proposed for finding the maximum clique in a graph. The proposed algorithm is tested on the DIMACS and BHOSLIB benchmark instances. The computational results are compared with similar literature researches. The results prove the effectiveness of proposed algorithm. © 2020, Springer Nature Switzerland AG