We study the Dominating set problem and Independent Set Problem for dynamic graphs in the vertex-arrival model. We say that a dynamic algorithm for one of these problems is $k$-stable when it makes at most $k$ changes to its output independent set or dominating set upon the arrival of each vertex. We study trade-offs between the stability parameter $k$ of the algorithm and the approximation ratio it achieves. We obtain the following results. 1. We show that there is a constant $\varepsilon^*>0$ such that any dynamic $(1+\varepsilon^*)$-approximation algorithm the for Dominating set problem has stability parameter $\Omega(n)$, even for bipartite graphs of maximum degree 4. 2. We present algorithms with very small stability parameters for the Dominating set problem in the setting where the arrival degree of each vertex is upper bounded by $d$. In particular, we give a $1$-stable $(d+1)^2$-approximation algorithm, a $3$-stable $(9d/2)$-approximation algorithm, and an $O(d)$-stable $O(1)$-approximation algorithm. 3. We show that there is a constant $\varepsilon^*>0$ such that any dynamic $(1+\varepsilon^*)$-approximation algorithm for the Independent Set Problem has stability parameter $\Omega(n)$, even for bipartite graphs of maximum degree $3$. 4. Finally, we present a $2$-stable $O(d)$-approximation algorithm for the Independent Set Problem, in the setting where the average degree of the graph is upper bounded by some constant $d$ at all times. We extend this latter algorithm to the fully dynamic model where vertices can also be deleted, achieving a $6$-stable $O(d)$-approximation algorithm.

翻译：本文研究了顶点到达模型中动态图的支配集问题与独立集问题。若一个针对此类问题的动态算法在每次顶点到达时，对其输出的独立集或支配集至多进行 $k$ 次修改，则称该算法为 $k$-稳定的。我们研究了算法的稳定性参数 $k$ 与其所达到的逼近比之间的权衡关系，并取得以下结果：1. 我们证明存在常数 $\varepsilon^*>0$，使得任何针对支配集问题的动态 $(1+\varepsilon^*)$-逼近算法都具有 $\Omega(n)$ 的稳定性参数，即使对于最大度为 4 的二部图亦然。2. 我们在每个顶点的到达度上界为 $d$ 的设置下，为支配集问题提出了具有极小稳定性参数的算法。具体而言，我们给出了一个 $1$-稳定的 $(d+1)^2$-逼近算法、一个 $3$-稳定的 $(9d/2)$-逼近算法以及一个 $O(d)$-稳定的 $O(1)$-逼近算法。3. 我们证明存在常数 $\varepsilon^*>0$，使得任何针对独立集问题的动态 $(1+\varepsilon^*)$-逼近算法都具有 $\Omega(n)$ 的稳定性参数，即使对于最大度为 $3$ 的二部图亦然。4. 最后，我们针对独立集问题提出了一个 $2$-稳定的 $O(d)$-逼近算法，该算法适用于图的平均度始终以某常数 $d$ 为上界的情形。我们将后一算法扩展至顶点亦可被删除的全动态模型，得到了一个 $6$-稳定的 $O(d)$-逼近算法。