Colorful Strips

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k−1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k+ ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k−1)+1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy. ∗Université Libre de Bruxelles, Brussels, Belgium. {galoupis, jcardin, secollet, mkormanc, slanger, ptaslaki}@ulb.ac.be. Partially supported by the Communauté française de Belgique ARC. †Chargé de Recherches du FRS-FNRS. ‡Nagoya University, Nagoya, Japan [email protected] §Mâıtre de Recherches du FRS-FNRS. ¶The Weizmann Institute of Science, Rehovot, Israel, [email protected] ‖Ben-Gurion University, Be’er Sheva, Israel. [email protected] 1 ar X iv :0 90 4. 21 15 v2 [ cs .C G ] 7 A pr 2 01 1