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- Han, Xin, et al.
(author)
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Approximating the maximum independent set and minimum vertex coloring on box graphs
- 2007
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In: Algorithmic Aspects in Information and Management / Lecture Notes in Computer Science. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 9783540728689 ; 4508, s. 337-345
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Conference paper (peer-reviewed)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 − 1n) factor from the size of its maximum clique and obtain an O(logd − 1n) approximation algorithm for minimum vertex coloring of such an intersection graph.
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