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