Evolutionary algorithms have been frequently used to deal with dynamic optimization problems, but their success is hard to understand from a theoretical perspective. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for dynamic combinatorial optimization problems. We examine a dynamic version of the classical vertex cover problem and analyse evolutionary algorithms with respect to their ability to maintain a 2-approximation. Analysing the different evolutionary algorithms studied by Jansen et al.[7], we point out where two previously studied approaches are not able to maintain a 2-approximation even if they start with a solution of that quality. Furthermore, we point out that the third approach is very effective in maintaining 2-approximations for the dynamic vertex cover problem.
Bi-level optimisation problems have gained increasing interest in the field of combinatorial optimisation in recent years. In this paper, we analyse the runtime of some evolutionary algorithms for bi-level optimisation problems. We examine two NP-hard problems, the generalised minimum spanning tree problem and the generalised travelling salesperson problem in the context of parameterised complexity. For the generalised minimum spanning tree problem, we analyse the two approaches presented by Hu and Raidl ( 2012 ) with respect to the number of clusters that distinguish each other by the chosen representation of possible solutions. Our results show that a (1+1) evolutionary algorithm working with the spanning nodes representation is not a fixed-parameter evolutionary algorithm for the problem, whereas the problem can be solved in fixed-parameter time with the global structure representation. We present hard instances for each approach and show that the two approaches are highly complementary by proving that they solve each other’s hard instances very efficiently. For the generalised travelling salesperson problem, we analyse the problem with respect to the number of clusters in the problem instance. Our results show that a (1+1) evolutionary algorithm working with the global structure representation is a fixed-parameter evolutionary algorithm for the problem.
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