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Abstract:
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.
