Parameterized Analysis of Multi-objective Evolutionary Algorithms and the Weighted Vertex Cover Problem

TL;DR

This paper extends the analysis of evolutionary algorithms to the weighted vertex cover problem, proposing algorithms with fixed parameter and approximation guarantees, including polynomial and near-optimal solutions.

Contribution

It introduces a fixed parameter evolutionary algorithm for weighted vertex cover and develops multi-objective algorithms achieving approximation ratios in expected polynomial time.

Findings

A fixed parameter evolutionary algorithm with respect to OPT.

A multi-objective evolutionary algorithm finds a 2-approximation in polynomial time.

A population-based algorithm achieves a (1+ε)-approximation with specified expected runtime.

Abstract

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 $O(n\cdot 2^{\min \{n,2(1- \varepsilon)OPT \}} + n^3)$.