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).

中文翻译：

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几何交点图的最大二部子图

我们研究最大二部子图 (MBS) 问题，其定义如下。给定平面中 $n$ 个几何对象的集合 $S$，我们想要计算最大大小的子集 $S'\subseteq S$，使得 $S'$ 中对象的交集图是二部的。我们首先给出一个简单的 $O(n)$-time 算法，它在一组 $n$ 间隔上解决 MBS 问题。我们还给出了一个 $O(n^2)$-time 算法，该算法可以为圆弧图上的问题计算接近最优的解决方案。我们证明 MBS 问题在几何图上是 NP 难的，其中最大独立集是 NP 难的（因此，即使在单位正方形和单位圆盘上也是 NP 难）。另一方面，我们针对单位正方形和单位圆盘上的问题给出了 PTAS。此外，我们展示了用于单位平方问题的小常数因子的快速逼近算法，单位圆盘和单位高度矩形。最后，我们研究了一个密切相关的几何问题，称为最大无三角形子图（TFS），其目标与 MBS 的目标相同，只是由集合 $S'$ 诱导的交集图只需要无三角形（而不是二分法）。