PTAS for Minimum Cost MultiCovering with Disks

SIAM Journal on Computing, Volume 53, Issue 4, Page 1181-1215, August 2024.
Abstract. In this paper, we study the following Minimum Cost Multicovering (MCMC) problem: Given a set of [math] client points [math] and a set of [math] server points [math] in a fixed dimensional [math] space, determine a set of disks centered at these server points so that each client point [math] is covered by at least [math] disks and the total cost of these disks is minimized, where [math] is a function that maps every client point to some nonnegative integer no more than [math] and the cost of each disk is measured by the [math]th power of its radius for some constant [math]. MCMC is a fundamental optimization problem with applications in many areas such as wireless/sensor networking. Despite extensive research on this problem for about two decades, only constant approximations were known for general [math]. It has been a long standing open problem to determine whether a PTAS is possible. In this paper, we give an affirmative answer to this question by presenting the first PTAS for it. Our approach is based on a number of novel techniques, such as balanced recursive realization and bubble charging, and new counterintuitive insights to the problem. Particularly, we approximate each disk with a set of sub-boxes and optimize them at the subdisk level. This allows us to first compute an approximate disk cover through dynamic programming, and then obtain the desired disk cover through a balanced recursive realization procedure.