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)


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.


Polynomial Time Geographic Information System Chromatic Number Maximum Clique Intersection Graph 
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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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