Complexity and approximation of the Smallest k-Enclosing Ball problem

Given an n-point set in a d-dimensional space and an integer k, consider the problem of finding a smallest ball enclosing at least k of the points. In the case of a fixed dimension, the problem is polynomial-time solvable but in general, when d is not fixed, the complexity status is not known. We prove the strong NP-hardness in the case of Euclidean space and the (2−ε)-inapproximability in the case of L∞ metric. We also present a simple 2-approximation algorithm for any metric, and PTAS for L∞-space of dimension O(logn).