Conflict-free colorings of rectangles ranges

Given the range space (P, R), where P is a set of n points in IR ² and R 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 (P, R) is conflict-free. We study the following question: Given P, is it possible to add a small set of points Q such that (P ∪ Q, R) can be colored with fewer colors than (P, R)? Our main result is the following: given P, and any ε ≥ 0, one can always add a set Q of O(n^1-ε) points such that P ∪ Q can be conflict-free colored using Õ(n^3/8(1+ε))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 recoloring technique, which we call quasi-conflict-free coloring, and which may be of independent interest. A further application of this technique is also given.