In this paper, we perform theoretical analyses on the behaviour of an evolutionary algorithm and a randomised search algorithm for the dynamic vertex cover problem based on its dual formulation. The dynamic vertex cover problem has already been theoretically investigated to some extent and it has been shown that using its dual formulation to represent possible solutions can lead to a better approximation behaviour. We improve some of the existing results, i.e. we find a linear expected re-optimization time for a ($1+1$) EA to re-discover a 2-approximation when edges are dynamically deleted from the graph. Furthermore, we investigate a different setting for applying the dynamism to the problem, in which a dynamic change happens at each step with a probability $P_D$. We also expand these analyses to the weighted vertex cover problem, in which weights are assigned to vertices and the goal is to find a cover set with minimum total weight. Similar to the classical case, the dynamic changes that we consider on the weighted vertex cover problem are adding and removing edges to and from the graph. We aim at finding a maximal solution for the dual problem, which gives a 2-approximate solution for the vertex cover problem. This is equivalent to the maximal matching problem for the classical vertex cover problem.

本文针对动态顶点覆盖问题的对偶表达式，对演化算法和随机搜索算法的行为进行了理论分析。理论上已经对动态顶点覆盖问题进行了一定程度的研究，结果表明，使用对偶表示来表示可能的解决方案可以带来更好的近似行为。我们改善了一些现有的结果，即我们找到了（）的线性预期重新优化时间$\mathrm{1个}+\mathrm{1个}$）当从图中动态删除边时，EA重新发现2近似值。此外，我们研究了将动态性应用于问题的不同设置，其中动态变化在每个步骤中的发生概率$P_d$。我们还将这些分析扩展到加权顶点覆盖问题，其中将权重分配给顶点，目标是找到具有最小总权重的覆盖集。与经典情况类似，我们在加权顶点覆盖问题上考虑的动态变化是在图上添加和删除边。我们旨在找到对偶问题的最大解，从而为顶点覆盖问题提供2个近似解。这等效于经典顶点覆盖问题的最大匹配问题。