Maximum Bipartite Subgraphs of Geometric Intersection Graphs

Article Type

Research Article

Publication Title

International Journal of Computational Geometry and Applications

Abstract

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 such that the intersection graph of the objects in S′ is bipartite. We first give an O(n2)-time algorithm that computes an almost 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 axis parallel rectangles. Additionally, we prove that the Maximum Triangle-free Subgraph (MTFS) problem is NP-hard for axis-parallel rectangles. Here the objective is the same as that of the MBS except the intersection graph induced by the set S′ needs to be triangle-free only (instead of being bipartite).

First Page

133

Last Page

157

DOI

https://10.1142/S021819592350005X

Publication Date

12-1-2023

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