Evolutionary multiobjective optimization for the classical vertex cover problem has been analysed in Kratsch and Neumann (2013) in the context of parameterized complexity analysis. This article extends the analysis to the weighted vertex cover problem in which integer weights are assigned to the vertices and the goal is to find a vertex cover of minimum weight. Using an alternative mutation operator introduced in Kratsch and Neumann (2013), we provide a fixed parameter evolutionary algorithm with respect to OPT, the cost of an optimal solution for the problem. Moreover, we present a multiobjective evolutionary algorithm with standard mutation operator that keeps the population size in a polynomial order by means of a proper diversity mechanism, and therefore, manages to find a 2-approximation in expected polynomial time. We also introduce a population-based evolutionary algorithm which finds a (1+ɛ)-approximation in expected time O(n·2min{n,2(1-ɛ)OPT}+n3).

中文翻译：

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多目标进化算法的参数化分析和加权顶点覆盖问题

Kratsch 和 Neumann (2013) 在参数化复杂性分析的背景下分析了经典顶点覆盖问题的进化多目标优化。本文将分析扩展到加权顶点覆盖问题，其中为顶点分配整数权重，目标是找到权重最小的顶点覆盖。使用 Kratsch 和 Neumann (2013) 中引入的替代变异算子，我们提供了一个关于 OPT 的固定参数进化算法，OPT 是问题的最佳解决方案的成本。此外，我们提出了一种具有标准变异算子的多目标进化算法，该算法通过适当的多样性机制将种群大小保持在多项式顺序，因此，设法在预期的多项式时间内找到 2-近似值。