A (1+1) Adaptive Memetic Algorithm for the Maximum Clique Problem

A memetic algorithm (MA) is an Evolutionary Algorithm (EA) augmented with a local search. We previously defined a (1+1) Adaptive Memetic Algorithm (AMA) with two different local searches, and the comparison with the well-known (1+1) EA, Dynamic (1+1) EA and (1+1) MA on some toy functions showed promise for our proposed algorithm. In this paper we focus on the NP-hard Maximum Clique Problem, and show the success of our proposed (1+1) AMA. We propose a new metric (expected running time to escape a local optimal), and show how this metric dominates the expected running time of finding a maximum clique. Then based on this new metric, we show the above analyzed algorithms are expected to find a maximum clique on graphs, bipartite graphs and sparse random graphs in a polynomial time in the number of vertices. Also based on our new metric, we will show that if an algorithm takes an exponential time to find a maximum clique of a graph, it must have been trapped into at least one local optimal that is extremely hard to escape. Furthermore, we will show that our proposed (1+1) AMA with a random permutation local search is expected to escape these (hard to escape) local optimal cliques drastically faster than the well-known basic (1+1) EA. The success of our experimental results also shows the benefit of our adaptive strategy combined with the random permutation local search.