Improved Bounds for Metric Capacitated Covering Problems

Abstract

In the Metric Capacitated Covering (MCC) problem, given a set of balls ${\mathcal {B}}$ in a metric space P with metric d and a capacity parameter U, the goal is to find a minimum sized subset ${{\mathcal {B}}}'\subseteq {\mathcal {B}}$ and an assignment of the points in P to the balls in ${\mathcal {B}}'$ such that each point is assigned to a ball that contains it and each ball is assigned with at most U points. MCC achieves an $O(\log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(\log |P|)$ even with $\beta < 3$ factor expansion of the balls. Bandyapadhyay et al. [Discrete and Computational Geometry 2019] showed that one can obtain an O(1)-approximation for the problem with 6.47 factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound 3 and the upper bound 6.47. In this current work, we show that it is possible to obtain an O(1)-approximation with only 4.24 factor expansion of the balls. Moreover, we show a similar upper bound of 5 for a more generalized version of MCC for which the best previously known bound was 9. We also study a closely related problem where instead of the upper bound, one needs to satisfy a lower bound on the number of points assigned to each ball in the solution. For this problem, we give an exact algorithm with only 5.83 factor expansion of the balls. All of our algorithms are based on LP rounding schemes that heavily exploit structure of fractional optimal solution.