PTAS FOR MINIMUM COST MULTICOVERING WITH DISKS

In this paper, we study the following Minimum Cost Multicovering (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 nonnegative integer no more than m and the cost of each disk is measured by the αth 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 for about two decades, only constant approximations were known for general k. 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.