Parameterized Analysis of Multi-objective Evolutionary Algorithms and the Weighted Vertex Cover Problem
This work addresses the weighted vertex cover problem for researchers in evolutionary computation and parameterized complexity, offering incremental extensions to prior analyses.
The paper tackles the weighted vertex cover problem by extending parameterized runtime analysis to it, presenting a fixed-parameter evolutionary algorithm with respect to OPT and achieving a 2-approximation in expected polynomial time and a (1+ε)-approximation in expected time O(n·2^{min{n,2(1-ε)OPT}} + n^3).
A rigorous runtime analysis of evolutionary multi-objective optimization for the classical vertex cover problem in the context of parameterized complexity analysis has been presented by Kratsch and Neumann (2013). In this paper, we extend the analysis to the weighted vertex cover problem and provide a fixed parameter evolutionary algorithm with respect to OPT, the cost of the the optimal solution for the problem. Moreover, using a diversity mechanisms, we present a multi-objective evolutionary algorithm that finds a 2-approximation in expected polynomial time and introduce a population-based evolutionary algorithm which finds a $(1+\varepsilon)$-approximation in expected time $O(n\cdot 2^{\min \{n,2(1- \varepsilon)OPT \}} + n^3)$.