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Parameterized analysis provides powerful mechanisms for obtainingfine-grained insights into different types of algorithms. In this work, wecombine this field with evolutionary algorithms and provide parameterizedcomplexity analysis of evolutionary multi-objective algorithms for the$W$-separator problem, which is a natural generalization of the vertex coverproblem. The goal is to remove the minimum number of vertices such that eachconnected component in the resulting graph has at most $W$ vertices. We providedifferent multi-objective formulations involving two or three objectives thatprovably lead to fixed-parameter evolutionary algorithms with respect to thevalue of an optimal solution $OPT$ and $W$. Of particular interest arekernelizations and the reducible structures used for them. We show that inexpectation the algorithms make incremental progress in finding such structuresand beyond. The current best known kernelization of the $W$-separator useslinear programming methods and requires a non-trivial post-process to extractthe reducible structures. We provide additional structural features to showthat evolutionary algorithms with appropriate objectives are also capable ofextracting them. Our results show that evolutionary algorithms with differentobjectives guide the search and admit fixed parameterized runtimes to solve orapproximate (even arbitrarily close) the $W$-separator problem.

Fixed Parameter Multi-Objective Evolutionary Algorithms for the W-Separator Problem