| 1. |
- 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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| 2. |
- Iwama, Kazuo, et al.
(author)
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Max-stretch reduction for tree spanners
- 2008
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In: Algorithmica. - : Springer Science and Business Media LLC. - 0178-4617 .- 1432-0541. ; 50:2, s. 223-235
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Journal article (peer-reviewed)abstract
- A tree t-spanner T of a graph G is a spanning tree of G whose max-stretch is t, i.e., the distance between any two vertices in T is at most t times their distance in G. If G has a tree t-spanner but not a tree (t−1)-spanner, then G is said to have max-stretch of t. In this paper, we study the Max-Stretch Reduction Problem: for an unweighted graph G=(V,E), find a set of edges not in E originally whose insertion into G can decrease the max-stretch of G. Our results are as follows: (i) For a ring graph, we give a linear-time algorithm which inserts k edges improving the max-stretch optimally. (ii) For a grid graph, we give a nearly optimal max-stretch reduction algorithm which preserves the structure of the grid. (iii) In the general case, we show that it is $mathcal{NP}$ -hard to decide, for a given graph G and its spanning tree of max-stretch t, whether or not one-edge insertion can decrease the max-stretch to t−1. (iv) Finally, we show that the max-stretch of an arbitrary graph on n vertices can be reduced to s′≥2 by inserting O(n/s′) edges, which can be determined in linear time, and observe that this number of edges is optimal up to a constant.
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