WALCOM 2020: WALCOM: Algorithms and Computation pp 158-169

# Maximum Bipartite Subgraph of Geometric Intersection Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12049)

## 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 show that the $$\texttt {MBS}$$ problem is $$\texttt {NP}$$-hard on geometric graphs for which the maximum independent set is $$\texttt {NP}$$-hard (hence, it is $$\texttt {NP}$$-hard even on unit squares and unit disks). On the algorithmic side, we first give a simple $$\mathcal {O}(n)$$-time algorithm that solves the $$\texttt {MBS}$$ problem on a set of n intervals. Then, we give an $$\mathcal {O}(n^2)$$-time algorithm that computes a near-optimal solution for the problem on circular-arc graphs. Moreover, for the approximability of the problem, we first present a $$\texttt {PTAS}$$ for the problem on unit squares and unit disks. Then, we present efficient 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 ($${{\texttt {\textit{MTFS}}}}$$), where the objective is the same as that of $$\texttt {MBS}$$ except the intersection graph induced by the set $$S'$$ needs to be triangle-free only (instead of being bipartite).

## Keywords

Bipartite subgraph Geometric intersection graphs $$\texttt {NP}$$-hardness Approximation schemes Triangle-free subgraph

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