Conflict-Free Colorings of Rectangle Ranges

Given the range space , where P is a set of n points in and is the family of subsets of P induced by all axis-parallel rectangles, the conflict-free coloring problem asks for a coloring of P with the minimum number of colors such that is conflict-free. We study the following question: Given P, is it possible to add a small set of points Q such that can be colored with fewer colors than ? Our main result is the following: given P, and any , one can always add a set Q of points such that P ∪ Q can be conflict-free colored using 1 colors. Moreover, the set Q and the conflict-free coloring can be computed in polynomial time, with high probability. Our result is obtained by introducing a general probabilistic re-coloring technique, which we call quasi-conflict-free coloring, and which may be of independent interest. A further application of this technique is also given