Near-linear-time algorithm for the geodetic Radon number of grids

The Radon number of a graph is the minimum integer r such that all sets of at least r of its vertices can be partitioned into two subsets whose convex hulls intersect. Determining the Radon number of general graphs in the geodetic convexity is NP-hard. In this paper, we show the problem is polynomia...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 210; pp. 277 - 283
Main Authors: Dourado, Mitre Costa, Pereira de Sá, Vinícius Gusmão, Rautenbach, Dieter, Szwarcfiter, Jayme Luiz
Format: Journal Article
Language:English
Published: Elsevier B.V 10.09.2016
Subjects:
Algorithms
Convexity
Geodetic convexity
Graph convexity
Graph theory
Graphs
Grids
Hulls (structures)
Mathematical analysis
Polynomials
Radon
Radon number
Grids
Graph convexity
Radon number
Geodetic convexity
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary:The Radon number of a graph is the minimum integer r such that all sets of at least r of its vertices can be partitioned into two subsets whose convex hulls intersect. Determining the Radon number of general graphs in the geodetic convexity is NP-hard. In this paper, we show the problem is polynomial for d-dimensional grids, for all d≥1. The proposed algorithm runs in near-linear O(d(logd)1/2) time for grids of arbitrary sizes, and in sub-linear O(logd) time when all grid dimensions have the same size.
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ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2015.05.001