Maintaining 2-Approximations for the Dynamic Vertex Cover Problem Using Evolutionary Algorithms

Abstract

Evolutionary algorithms have been frequently used to deal with dynamic optimization problems, but their success is hard to understand from a theoretical perspective. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for dynamic combinatorial optimization problems. We examine a dynamic version of the classical vertex cover problem and analyse evolutionary algorithms with respect to their ability to maintain a 2-approximation. Analysing the different evolutionary algorithms studied by Jansen et al.[7], we point out where two previously studied approaches are not able to maintain a 2-approximation even if they start with a solution of that quality. Furthermore, we point out that the third approach is very effective in maintaining 2-approximations for the dynamic vertex cover problem.

Runtime analysis of RLS and (1 + 1) EA for the dynamic weighted vertex cover problem

“…A dynamic change affects the graph by either deleting an edge or adding it. However, it is already shown that (1+1) EA e restores the quality of 2-approximation when a new edge is added dynamically in expected time O(m) [16]. In Theorem 8 we prove that the expected re-optimisation time of (1+1) EA e after a dynamic deletion is also O(m).…”

“…In this paper, we have investigated the behaviour of two simple randomised search heuristics, namely RLS and (1+1) EA, on the dynamic vertex cover problem. The expected required time for re-optimization of the classical vertex cover problem after a dynamic change had been analysed in [16] for these two algorithms with an edge-based representation. In this paper we have improved the results of the analysis of (1+1) EA.…”

“…The second part is a penalty for edges that solution s does not cover, and the third part is an extra penalty inspired from the fact that a maximal matching induces a 2-approximate solution for the vertex cover problem. Algorithm 1 Edge-Based (1+1) EA ((1+1) EA e ) [16] 1: The initial solution, s, is given: a bit-string of size m which used to be a 2-approximate solution before changing the graph; 2: while Stopping criteria not met do 3: Set s := s;…”

“…We investigate the behaviour of evolutionary algorithms when this constraint changes through the addition and removal of edges. In [16] the vertex cover problem is considered with a simple dynamic setting where the rate of dynamic changes is small enough, so that the studied algorithms can re-optimize the problem after a dynamic change, before the following change happens. We call this dynamic setting One-time Dynamic Setting.…”

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

“…(More details of the LP formulation and its dual formulation for the fractional WVC can be found in the next section. ) Pourhassan et al [22] presented a dynamic model of the Vertex Cover problem, in which the graph-editing operation adds (resp., removes) exactly one edge into (resp., from) the given unweighted graph, and analyzed evolutionary algorithms with respect to their abilities to maintain a 2-approximate solution. They examined different variants of the RLS and (1+1) EA, node-based representation and edge-based representation.…”