Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm NP}$-complete and ${\rm APX}$-hard, and the best approximation ratio known is $\Omega\left({\rm OPT}^{-1/2}\right) =\Omega(n^{-1/2})$. Here we obtain better approximations in two specials cases: (I) A constant factor approximation for the case where the input set consists of lattice points and is \emph{dense}, which means that the ratio between the maximum and the minimum distance in $P$ is of the order of $\Theta(\sqrt{n})$. (II) An $\Omega\left((\log{n})^{-1/2}\right)$-approximation for the case where the input set is the set of vertices of a \emph{generic} $n$-line arrangement, i.e., one with $\Omega(n^2)$ vertices. The scenario in (I) is a special case of that in (II). Our approximations rely on probabilistic methods and results from incidence geometry.

翻译：给定平面上一个点集，\textsc{一般位置子集选择}问题旨在寻找一个最大规模的子集，使得其中任意三点不共线。该问题已知是 ${\rm NP}$ 完全且 ${\rm APX}$ 难的，目前已知的最佳近似比为 $\Omega\left({\rm OPT}^{-1/2}\right) =\Omega(n^{-1/2})$。本文在两种特殊情况下获得了更好的近似结果：（I）当输入点集由格点构成且是\emph{稠密}的（即点集 $P$ 中最大距离与最小距离之比为 $\Theta(\sqrt{n})$ 量级）时，给出常数因子近似算法。（II）当输入点集是\emph{一般} $n$ 直线排列（即具有 $\Omega(n^2)$ 个顶点）的顶点集时，给出 $\Omega\left((\log{n})^{-1/2}\right)$ 近似算法。情况（I）是情况（II）的特例。我们的近似方法依赖于概率论技巧与几何关联理论中的结果。