Covering points by disjoint boxes with outliers

For a set of n points in the plane, we consider the axis–aligned (p, k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain at least n − k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n + k log k) time for p = 1, and O(n log n + kp logp k) time for p = 2, 3. For rectangles we get O(n + k) for p = 1 and O(n log n+ k log k) time for p = 2, 3. In all cases, our algorithms use O(n) space.