Maximum Bipartite Subgraph of Geometric Intersection Graphs

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).