Covering Points by Disjoint Boxes with Outliers

TL;DR

This paper investigates the NP-hard problem of covering points with disjoint axis-aligned boxes, providing efficient algorithms for small fixed numbers of boxes and analyzing the problem's computational complexity.

Contribution

It introduces algorithms for covering points with disjoint squares or rectangles for small fixed p, and proves NP-hardness for general p.

Findings

NP-hardness for general p

Efficient algorithms for p=1,2,3 with specific running times

Algorithms use linear space

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 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.