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/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 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(logd − 1 n) factor from the size of its maximum clique and obtain an O(logd − 1 n) approximation algorithm for minimum vertex coloring of such an intersection graph.
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Han, X., Iwama, K., Klein, R., Lingas, A. (2007). Approximating the Maximum Independent Set and Minimum Vertex Coloring on Box Graphs. In: Kao, MY., Li, XY. (eds) Algorithmic Aspects in Information and Management. AAIM 2007. Lecture Notes in Computer Science, vol 4508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72870-2_32
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DOI: https://doi.org/10.1007/978-3-540-72870-2_32
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