New results on the coarseness of bicolored point sets

Let $S$ be a 2-colored (red and blue) set of $n$ points in the plane. A subset $I$ of $S$ is an island if there exits a convex set $C$ such that $I=C\cap S$. The discrepancy of an island is the absolute value of the number of red minus the number of blue points it contains. A convex partition of $S$ is a partition of $S$ into islands with pairwise disjoint convex hulls. The discrepancy of a convex partition is the discrepancy of its island of minimum discrepancy. The coarseness of $S$ is the discrepancy of the convex partition of $S$ with maximum discrepancy. This concept was recently defined by Bereg et al. [CGTA 2013]. In this paper we study the following problem: Given a set $S$ of $n$ points in general position in the plane, how to color each of them (red or blue) such that the resulting 2-colored point set has small coarseness? We prove that every $n$-point set $S$ can be colored such that its coarseness is $O(n^{{1/4}\sqrt{\log} n})$. This bound is almost tight since there exist $n$-point sets such that every 2-coloring gives coarseness at least $\Omega(n^{1/4})$. Additionally, we show that there exists an approximation algorithm for computing the coarseness of a 2-colored point set, whose ratio is between $1/128$ and $1/64$, solving an open problem posted by Bereg et al. [CGTA 2013]. All our results consider $k$-separable islands of $S$, for some $k$, which are those resulting from intersecting $S$ with at most $k$ halfplanes.