# The container selection problem

## Abstract

We introduce and study a network resource management problem that is a special case of nonmetric k-median, naturally arising in cross platform scheduling and cloud computing. In the continuous d-dimensional container selection problem, we are given a set C ⊂ Rd of input points, for some d ≥ 2, and a budget k. An input point p can be assigned to a "container point" c only if c dominates p in every dimension. The assignment cost is then equal to the ell;1-norm of the container point. The goal is to find k container points in Rd, such that the total assignment cost for all input points is minimized. The discrete variant of the problem has one key distinction, namely, the container points must be chosen from a given set F of points. For the continuous version, we obtain a polynomial time approximation scheme for any fixed dimension d ≥ 2. On the negative side, we show that the problem is NP-hard for any d ≥ 3. We further show that the discrete version is significantly harder, as it is NP-hard to approximate without violating the budget k in any dimension d ≥ 3. Thus, we focus on obtaining bi-approximation algorithms. For d = 2, the bi-approximation guarantee is (1+ε, 3), i.e., for any ε > 0, our scheme outputs a solution of size 3k and cost at most (1 + ε) times the optimum. For fixed d > 2, we present a (1 + ε,O(1/ε log k)) bi-approximation algorithm.