General Position Subset Selection in Line Arrangements

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.