Strong Conflict-Free Coloring for Intervals

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 5-2/k when k ≥ 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 ≥ 1. We also provide, in case k = O(polylog(n)), a quasipolynomial time algorithm to decide the existence of a k-SCF coloring that uses at most q colors.