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

  • Authors:

  • Xin Han;Kazuo Iwama;Rolf Klein;Andrzej Lingas

  • Affiliations:

  • School of Informatics, Kyoto University, Kyoto 606-8501, Japan;School of Informatics, Kyoto University, Kyoto 606-8501, Japan;University of Bonn, Institute of Computer Science I, D-53117 Bonn, Germany;Department of Computer Science, Lund University, 221 00 Lund, Sweden

  • Venue:
  • AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
  • Year:
  • 2007

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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 nvertices which have an independent set of size 茂戮驴(n/logO(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 nvertices 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 nd-dimensional orthogonal rectangles is within an O(logd茂戮驴 1n) factor from the size of its maximum clique and obtain an O(logd茂戮驴 1n) approximation algorithm for minimum vertex coloring of such an intersection graph.