Maximum Bipartite Subgraph of Geometric Intersection Graphs

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


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).


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



We thank Michiel Smid for useful discussions on the problem.


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Authors and Affiliations

  1. 1.Indian Statistical InstituteKolkataIndia
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

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