Maximum Bipartite Subgraph of Geometric Intersection Graphs

TL;DR

This paper investigates algorithms and complexity for finding maximum bipartite subgraphs in geometric intersection graphs, providing efficient algorithms for specific cases and establishing NP-hardness and approximation results for general cases.

Contribution

It introduces new algorithms for the MBS problem on intervals and circular-arc graphs, proves NP-hardness on certain geometric graphs, and offers PTAS and approximation algorithms for unit squares and disks.

Findings

Linear-time algorithm for intervals

NP-hardness on unit squares and disks

PTAS and approximation algorithms for unit squares and disks

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 $S'\subseteq S$ 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 $O(n^2)$-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).