Covering points by disjoint boxes with outliers

Abstract

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 + k(p) log(p) k) time for p = 2, 3. For rectangles we get O (n + k(3)) for p = 1 and O (n log n + k(2+p) log(p-1) k) time for p = 2, 3. In all cases, our algorithms use O (n) space.