We study the k-center problem in a kinetic setting: given a set of continuously moving points P in the plane, determine a set of k (moving) disks that cover P at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model allows positive results only for $k<3$. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of k-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution—the topological stability ratio—considering various metrics and various optimization criteria. For $k=2$ we provide tight bounds, and for small $k>2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant k.

我们研究动力学设置中的k中心问题：给定平面中的一组连续移动点P，确定在每个时间步都覆盖P的一组k个（移动）磁盘，以使磁盘尽可能小在任何时间点。尽管随着时间的推移，最佳解决方案可能会出现不连续的变化，但是许多实际应用要求解决方案必须稳定：磁盘必须随着时间的推移平稳移动。关于此问题的现有结果要求磁盘以有限的速度移动，但是此模型仅对以下情况允许正面结果：$ķ<3$。因此，结果是有限的，几乎没有理论上的洞察力。相反，我们研究了拓扑的稳定的ķ -centers。拓扑稳定性是最近引入的，仅要求解决方案不断变化，但可能会快速变化。我们证明了最优但不稳定的解决方案的半径与拓扑稳定的解决方案的半径之比（拓扑稳定性比）的上限和下限，其中考虑了各种度量标准和各种优化标准。为了$ķ=\mathrm{2个}$ 我们提供了紧密的界限，对于小规模 $ķ>\mathrm{2个}$我们可以获得不重要的上下限。最后，我们提供了一种计算常数k的多项式时间内的拓扑稳定性比率的算法。