Fully-dynamic and kinetic conflict-free coloring of intervals with respect to points

We introduce the fully-dynamic conflict-free coloring problem for a set S of intervals in 1 with respect to points, where the goal is to maintain a conflict-free coloring for S under insertions and deletions. A coloring is conflict-free if for each point p contained in some interval, p is contained in an interval whose color is not shared with any other interval containing p. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is ω(log n/loglog n), and that any strategy using O(1/) colors needs ω(n) recolorings; a coloring strategy that uses O(log n) colors at the cost of O(log n) recolorings, and another strategy that uses O(1/) colors at the cost of O(n/) recolorings; stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.