Runtime Performances of Randomized Search Heuristics for the Dynamic Weighted Vertex Cover Problem

TL;DR

This paper analyzes the runtime efficiency of randomized search heuristics, specifically evolutionary algorithms, for maintaining approximate solutions to the dynamic weighted vertex cover problem under graph modifications.

Contribution

It provides a theoretical analysis of how adapted evolutionary algorithms perform in dynamic settings for the weighted vertex cover problem, including runtime bounds and the impact of step size adaptation.

Findings

Three algorithms maintain 2-approximate solutions in polynomial expected time.

The (1+1) EA with 1/5-th rule has pseudo-polynomial expected runtime.

Step size adaptation strategies influence algorithm efficiency in dynamic environments.

Abstract

Randomized search heuristics such as evolutionary algorithms are frequently applied to dynamic combinatorial optimization problems. Within this paper, we present a dynamic model of the classic Weighted Vertex Cover problem and analyze the runtime performances of the well-studied algorithms Randomized Local Search and (1+1) EA adapted to it, to contribute to the theoretical understanding of evolutionary computing for problems with dynamic changes. In our investigations, we use an edge-based representation based on the dual form of the Linear Programming formulation for the problem and study the expected runtime that the adapted algorithms require to maintain a 2-approximate solution when the given weighted graph is modified by an edge-editing or weight-editing operation. Considering the weights on the vertices may be exponentially large with respect to the size of the graph, the step size adaption strategy is incorporated, with or without the 1/5-th rule that is employed to control the increasing/decreasing rate of the step size. Our results show that three of the four algorithms presented in the paper can recompute 2-approximate solutions for the studied dynamic changes in polynomial expected runtime, but the (1+1) EA with 1/5-th Rule requires pseudo-polynomial expected runtime.