TY - GEN

T1 - Colorful strips

AU - Aloupis, Greg

AU - Cardinal, Jean

AU - Collette, Sébastien

AU - Imahori, Shinji

AU - Korman, Matias

AU - Langerman, Stefan

AU - Schwartz, Oded

AU - Smorodinsky, Shakhar

AU - Taslakian, Perouz

PY - 2010

Y1 - 2010

N2 - 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(4ln 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. Lower bounds are also given for all of the above problems. 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.

AB - 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(4ln 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. Lower bounds are also given for all of the above problems. 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.

UR - http://www.scopus.com/inward/record.url?scp=77953487297&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-12200-2_2

DO - 10.1007/978-3-642-12200-2_2

M3 - Conference contribution

AN - SCOPUS:77953487297

SN - 3642121993

SN - 9783642121999

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 2

EP - 13

BT - LATIN 2010

T2 - 9th Latin American Theoretical Informatics Symposium, LATIN 2010

Y2 - 19 April 2010 through 23 April 2010

ER -