Strong Conflict-Free Coloring for Intervals

Abstract

We consider the $k$-strong conflict-free ($k$-SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval $I$ of the family there are at least $k$ colors each appearing exactly once in $I$. We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when $k=1$ and $5-\frac{2}{k}$ when $k\ge 2$. In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any $k \ge 1$. We also provide, in case $k=O({{\mathrm{polylog}}}(n))$, a quasipolynomial time algorithm to decide the existence of a $k$-SCF coloring that uses at most $q$ colors.