We study the Maximum Bipartite Subgraph (MBS) problem, which is defined as follows. Given a set S of n geometric objects in the plane, we want to compute a maximum-size subset S′⊆S such that the intersection graph of the objects in S′ is bipartite. We first give a simple O(n)-time algorithm that solves the MBS problem on a set of n intervals. We also give an O(n2)-time algorithm that computes a near-optimal solution for the problem on circular-arc graphs. We show that the MBS problem is NP-hard on geometric graphs for which the maximum independent set is NP-hard (hence, it is NP-hard even on unit squares and unit disks). On the other hand, we give a PTAS for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks and unit-height rectangles. Finally, we study a closely related geometric problem, called Maximum Triangle-free Subgraph (TFS), where the objective is the same as that of MBS except the intersection graph induced by the set S′ needs to be triangle-free only (instead of being bipartite).
@article{arxiv.1909.03896,
title = {Maximum Bipartite Subgraph of Geometric Intersection Graphs},
author = {Satyabrata Jana and Anil Maheshwari and Saeed Mehrabi and Sasanka Roy},
journal= {arXiv preprint arXiv:1909.03896},
year = {2020}
}