An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting
We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in R^3. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be "lifted" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.
Geometric permutation
Emptiness testing of semi-algebraic sets
Computer-aided proof
Theory of computation~Computational geometry
Computing methodologies~Combinatorial algorithms
40:1-40:16
Regular Paper
A full version of this paper, available at https://arxiv.org/abs/1903.03014, contains some details omitted from this version due to space constraints.
Xavier
Goaoc
Xavier Goaoc
Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Supported by Institut Universitaire de France.
Andreas
Holmsen
Andreas Holmsen
Department of Mathematical Sciences, KAIST, Daejeon, South Korea
Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03930998).
Cyril
Nicaud
Cyril Nicaud
Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE, UPEM, F-77454, Marne-la-Vallée, France
10.4230/LIPIcs.SoCG.2019.40
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Xavier Goaoc, Andreas Holmsen, and Cyril Nicaud
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