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 three 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). (III) An $\Omega\left((\log{n})^{-1/2}\right)$-approximation for the case where the input set has at most $O(\sqrt{n})$ points collinear and can be covered by $O(\sqrt{n})$ lines. Our approximations rely on probabilistic methods and results from incidence geometry.

翻译：给定平面上的一个点集，\textsc{一般位置子集选取}问题旨在找到一个最大规模的子集，使得其中任意三点不共线。已知该问题是${\rm NP}$完全且${\rm APX}$难的，目前已知的最佳近似比为$\Omega\left({\rm OPT}^{-1/2}\right) =\Omega(n^{-1/2})$。本文在三种特殊情形下获得了更好的近似结果：（I）当输入点集由格点构成且为\emph{稠密}集（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）当输入点集为\emph{一般性}$n$线构型（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。情形（I）是情形（II）的特例。（III）当输入点集中至多有$O(\sqrt{n})$个点共线且可被$O(\sqrt{n})$条直线覆盖时，给出$\Omega\left((\log{n})^{-1/2}\right)$近似算法。我们的近似算法基于概率方法及关联几何中的相关结果。