A branch-and-bound algorithm for the minimum radius $k$-enclosing ball problem

The minimum k-enclosing ball problem seeks the ball with smallest radius that contains at least k of m given points in a general n-dimensional Euclidean space. This problem is NP-hard. We present a branch-and-bound algorithm on the tree of the subsets of k points to solve this problem. The nodes on the tree are ordered in a suitable way, which, complemented with a last-in-first-out search strategy, allows for only a small fraction of nodes to be explored. Additionally, an efficient dual algorithm to solve the subproblems at each node is employed.