Separating Colored Points with Minimum Number of Rectangles

TL;DR

This paper investigates the problem of covering multiple disjoint point sets with the fewest rectangles, proving that finding an optimal solution is NP-hard, highlighting computational complexity challenges.

Contribution

It introduces a novel geometric covering problem and establishes its NP-hardness, contributing to computational geometry and complexity theory.

Findings

The problem is NP-hard.

Minimal rectangle cover is computationally challenging.

Provides theoretical foundation for future approximation algorithms.

Abstract

In this paper we study the following problem: Given $k$ disjoint sets of points, $P_1, \ldots, P_k$ on the plane, find a minimum cardinality set $\mathcal{T}$ of arbitrary rectangles such that each rectangle contains points of just one set $P_i$ but not the others. We prove the NP-hardness of this problem.