Runtime Analysis of a (1+1) Adaptive Memetic Algorithm and the Maximum Clique Problem

A memetic algorithm is an evolutionary algorithm (EA) augmented with a local search. For many applications, researchers have applied variations of memetic algorithms and have gained very positive experimental results. But the theory of these variations of memetic algorithms is still underdeveloped. This paper defines the (1+1) adaptive memetic algorithm (AMA) with a dynamic mutation probability, and analyzes two types of local searches. We then propose different classes of functions for studying the performance of evolutionary algorithms. We give time complexity analysis that proves our two local searches can outperform each other on different functions. Also we show that memetic algorithms with dynamic mutation probabilities can outperform memetic algorithms with static mutation probabilities, and vice versa. Then, 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 planar 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 which 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.