TY - GEN

T1 - Conict-free coloring of intersection graphs of geometric objects

AU - Keller, Chaya

AU - Smorodinsky, Shakhar

N1 - Funding Information:
∗Department of Mathematics, Ben-Gurion University of the NEGEV, Be’er-Sheva Israel. kellerc@math.bgu.ac.il. Research partially supported by Grant 635/16 from the Israel Science Foundation, by the Shulamit Aloni PostDoctoral Fellowship of the Israeli Ministry of Science and Technology, and by the Kreitman FoundationPost-Doctoral Fellowship.
Publisher Copyright:
© Copyright 2018 by SIAM.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In FOCS'2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflictfree colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with O(log n) colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we obtain the following strengthening of the two main results of Even et al.: Any family F of n discs in the plane can be colored with O(log n) colors in such a way that for any disc B in the plane, the set of discs in F that intersect B contains a uniquely-colored element. Moreover, such a coloring can be computed deterministically in polynomial time. In view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of ?-fat objects in the plane.

AB - In FOCS'2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflictfree colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with O(log n) colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we obtain the following strengthening of the two main results of Even et al.: Any family F of n discs in the plane can be colored with O(log n) colors in such a way that for any disc B in the plane, the set of discs in F that intersect B contains a uniquely-colored element. Moreover, such a coloring can be computed deterministically in polynomial time. In view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of ?-fat objects in the plane.

UR - http://www.scopus.com/inward/record.url?scp=85045546675&partnerID=8YFLogxK

U2 - 10.1137/1.9781611975031.154

DO - 10.1137/1.9781611975031.154

M3 - Conference contribution

AN - SCOPUS:85045546675

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 2397

EP - 2411

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

A2 - Czumaj, Artur

PB - Association for Computing Machinery

T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

Y2 - 7 January 2018 through 10 January 2018

ER -