Maximum Coverage by k Lines

Given a set of n disks in the plane, we study the problem of finding k lines that together intersect the maximum number of input disks. We consider three variants of this problem with the following constraints on the solution: (1) no constraint on the lines, (2) the k lines should be parallel and (3) the k lines should pass through a common point. For k=2, we give O(n3logn)-time algorithms for all three cases. For any fixed k≥3, we give an O(n3k/2)-time algorithm for (1). For variants (2) and (3), the running times of our algorithms vary from O(n4) to O(n6).