In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B' subseteq B of balls and assign each point in P to some ball in B' that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B'. We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only (1+epsilon) factor expansion is sufficient for any epsilon > 0, with the approximation factor being a polynomial in 1/epsilon. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.

Recently, Adamaszek and Wiese [1, 2] presented a quasi-polynomial time approximation scheme (QPTAS) for the problem of computing a maximum weight independent set for certain families of planar objects. This major advance on the problem was based on their proof that a certain type of separator exists for any independent set. Subsequently, Har-Peled [22] simplified and generalized their result. Mustafa et al. [36] also described a simplification, and somewhat surprisingly, showed that QPTAS's can be obtained for certain, albeit special, type of covering problems. Building on these developments, we revisit two NP-hard geometric partitioning problems – convex decomposition and surface approximation. Partitioning problems combine the features of packing and covering. In particular, since the optimal solution does form a packing, the separator theorems are potentially applicable. Nevertheless, the two partitioning problems we study bring up additional difficulties that are worth examining in the context of the wider applicability of the separator methodology. We show how these issues can be handled in presenting quasi-polynomial time algorithms for these two problems with improved approximation guarantees.

We consider variants of the following multi-covering problem with disks. We are given two point sets Y (servers) and X (clients) in the plane, and a coverage function κ :X -> N. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client x ∈ X be covered by at least κ(x) of the server disks. The objective function we wish to minimize is the sum of the areas of the disks. We present a polynomial time algorithm for this problem achieving an O(1) approximation.