Runtime analysis comparison of two fitness functions on a memetic algorithm for the Clique Problem

Abstract

It is commonly accepted that a proper fitness function can guide the algorithm to find a global optimum solution faster. This paper will use the runtime analysis to provide the theoretical evidence that a small change of the fitness function (additional one step looking forward) can result in a huge performance gap in terms of finding a global optimum solution. It also shows that the fitness function that gives the best results in an Memetic Algorithm on the Clique Problem is entirely instance specific. In detail, we will formalize a (1+1) Restart Memetic Algorithm with a Best-Improvement Local Search, and run them on two different fitness functions, fOL and fOPL, to solve the Clique Problem respectively. We then construct two families of graphs, G1 and G2, and show that, for the first family of graphs G1, the (1+1) RMA on the fitness function fOPL drastically outperforms the (1+1) RMA on the fitness function fOL, and vice versa for the second family of graphs G2.

Memetic algorithms outperform evolutionary algorithms in multimodal optimisation

“…Sudholt and Zarges [29] investigated the use of MAs for the graph colouring problem and showed that MAs can efficiently colour bipartite graphs with 2 colours and colour planar graphs with maximum degree up to 6 with up to 5 colours. Wei and Dinneen analysed MAs for solving the Clique problem, investigating the choice of the fitness function [30] as well as the choice of the local search operator [31]. They gave families of graphs where one choice of fitness function outperforms another and where the (1+1) MA with first-improvement local search (FILS) and the (1+1) MA with best-improvement local search (BILS) outperform one another.…”

Hybridizing the dynamic mutation approach with local searches to overcome local optima

“…For example, Sudholt, in 2006 [14] and in 2009 [16], showed that changing the depth of local searches or the frequency of applying local searches in MAs will reduce the performance from polynomial to super-polynomial. In 2014, Wei and Dinneen in [19] showed that the best local search in MAs for solving the Clique Problem is instance specific; also in [18], they showed that the best fitness function for MAs to solve the Clique Problem is instance specific. Furthermore, the interaction of mutations and local searches has attracted much attention, which is also the focus of this paper.…”