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Approximating the Maximum Independent Set and Minimum Vertex Coloring on Box Graphs

  • Xin Han
  • Kazuo Iwama
  • Rolf Klein
  • Andrzej Lingas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4508)

Abstract

A box graph is the intersection graph of a finite set of orthogonal rectangles in the plane. The problem of whether or not the maximum independent set problem (MIS for short) for box graphs can be approximated within a substantially sub-logarithmic factor in polynomial time has been open for several years. We show that for box graphs on n vertices which have an independent set of size Ω(n/log O(1) n) the maximum independent set problem can be approximated within O(logn / loglogn) in polynomial time. Furthermore, we show that the chromatic number of a box graph on n vertices is within an O(logn) factor from the size of its maximum clique and provide an O(logn) approximation algorithm for minimum vertex coloring of such a box graph. More generally, we can show that the chromatic number of the intersection graph of n d-dimensional orthogonal rectangles is within an O(log d − 1 n) factor from the size of its maximum clique and obtain an O(log d − 1 n) approximation algorithm for minimum vertex coloring of such an intersection graph.

Keywords

Polynomial Time Geographic Information System Chromatic Number Maximum Clique Intersection Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Xin Han
    • 1
  • Kazuo Iwama
    • 1
  • Rolf Klein
    • 2
  • Andrzej Lingas
    • 3
  1. 1.School of Informatics, Kyoto University, Kyoto 606-8501Japan
  2. 2.University of Bonn, Institute of Computer Science I, D-53117 BonnGermany
  3. 3.Department of Computer Science, Lund University, 221 00 LundSweden

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