848 research outputs found
Conflict-Free Coloring of Intersection Graphs of Geometric Objects
In FOCS'2002, Even et al. introduced and studied the notion of conflict-free
colorings of geometrically defined hypergraphs. They motivated it by frequency
assignment problems in cellular networks. This notion has been extensively
studied since then.
A conflict-free coloring of a graph is a coloring of its vertices such that
the neighborhood (pointed or closed) of each vertex contains a vertex whose
color differs from the colors of all other vertices in that neighborhood. In
this paper we study conflict-colorings of intersection graphs of geometric
objects. We show that any intersection graph of n pseudo-discs in the plane
admits a conflict-free coloring with O(\log n) colors, with respect to both
closed and pointed neighborhoods. We also show that the latter bound is
asymptotically sharp. Using our methods, we also obtain a strengthening of the
two main results of Even et al. which we believe is of independent interest. In
particular, in view of the original motivation to study such colorings, this
strengthening suggests further applications to frequency assignment in wireless
networks.
Finally, we present bounds on the number of colors needed for conflict-free
colorings of other classes of intersection graphs, including intersection
graphs of axis-parallel rectangles and of \rho-fat objects in the plane.Comment: 18 page
Stabbing line segments with disks: complexity and approximation algorithms
Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least cardinality
set of disks of fixed radii where the set of segments forms a straight
line drawing of a planar graph without edge crossings. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for and some constant where and
are Euclidean lengths of the longest and shortest graph edges
respectively. Fast -time -approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality holds uniformly for some constant
i.e. when lengths of edges of are uniformly bounded from above by
some linear function of Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Efficient NC algorithms for set cover with applications to learning and geometry
In this paper we give the first NC approximation algorithms for the unweighted and weighted set cover problems. Our algorithms use a linear number of processors and give a cover that has at most log n times the optimal size/weight, thus matching the performance of the best sequential algorithms. We apply our set cover algorithm to learning theory, giving an NC algorithm to learn the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC-dimension that has an NC hypothesis finder. In addition, we give a linear-processor NC algorithm for a variant of the set cover problem first proposed by Chazelle and Friedman and use it to obtain NC algorithms for several problems in computational geometry
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