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I. van der Hoog, M. van Kreveld, W. Meulemans, K. Verbeek, J. Wulms:
"Topological stability of kinetic k-centers";
Theoretical Computer Science, 866 (2021), 145 - 159.

We study the k-center problem in a kinetic setting: given a set of continuously moving points Pin the plane, determine a set of k(moving) disks that cover Pat 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 stabilityof 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 =2we provide tight bounds, and for small k >2we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant k.

https://publik.tuwien.ac.at/files/publik_300342.pdf