PTAS for minimum cost multi-covering with disks

In this paper, we study the following Minimum Cost Multi-Covering (MCMC) problem: Given a set of n client points C and a set of m server points S in a fixed dimensional R^d space, determine a set of disks centered at these server points so that each client point c is covered by at least k(c) disks and the total cost of these disks is minimized, where k(·) is a function that maps every client point to some non-negative integer no more than m and the cost of each disk is measured by the α-th power of its radius for some constant α > 0. MCMC is a fundamental optimization problem with applications in many areas such as wireless/sensor networking. Despite extensive research on this problem in the past two decades, only constant approximations were known for general k. It has been an open problem for a long time 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 insights to the problem which are somewhat counter-intuitive. Particularly, we show that instead of optimizing each disk as a whole, it is possible to further approximate each disk with a set of sub-boxes and optimize them at the sub-disk level. This allows us to first compute an approximate disk cover with minimum cost through dynamic programming, and then obtain the desired disk cover through a balanced recursive realization procedure. Our techniques have the potential to be used to other geometric (covering) problems.