Independent sets and hitting sets of bicolored rectangular families

TL;DR

This paper studies bicolored rectangular families, proving a key equality between maximum independent and minimum hitting sets, and provides polynomial algorithms for these problems along with applications to bipartite graph covers.

Contribution

It introduces polynomial algorithms for maximum independent sets and minimum hitting sets in bicolored rectangular families, extending solutions to related bipartite graph problems.

Findings

Maximum independent set equals minimum hitting set in BRFs.

Polynomial algorithms for biclique covers and cross-free matchings.

NP-hardness of maximum weighted independent set in weighted BRFs.

Abstract

A bicolored rectangular family BRF is a collection of all axis-parallel rectangles contained in a given region Z of the plane formed by selecting a bottom-left corner from a set A and an upper-right corner from a set B. We prove that the maximum independent set and the minimum hitting set of a BRF have the same cardinality and devise polynomial time algorithms to compute both. As a direct consequence, we obtain the first polynomial time algorithm to compute minimum biclique covers, maximum cross-free matchings and jump numbers in a class of bipartite graphs that significantly extends convex bipartite graphs and interval bigraphs. We also establish several connections between our work and other seemingly unrelated problems. Furthermore, when the bicolored rectangular family is weighted, we show that the problem of finding the maximum weight of an independent set is NP-hard, and provide efficient algorithms to solve it on certain subclasses.