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Combinatorial and Geometric View of the System Reliability Theory

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Book cover Mathematical Software – ICMS 2016 (ICMS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9725))

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Abstract

Associated to every coherent system there is a canonical ideal whose Hilbert series encodes the reliability of the system. We study various ideals arising in the theory of system reliability. Using ideas from the theory of orientations, and matroids on graphs we associate a polyhedral complex to our system so that the non-cancelling terms in the reliability formula can be read from the labeled faces of this complex. Algebraically, this polyhedron resolves the minimal free resolution of these ideals. In each case, we give an explicit combinatorial description of non-cancelling terms in terms of acyclic orientations of graph and the number of regions in the graphic hyperplane arrangement. This resolves open questions posed by Giglio-Wynn and develops new connections between the theory of oriented matroid, the theory of divisors on graphs, and the theory of system reliability.

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Fatemeh Mohammadi

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  1. Fatemeh Mohammadi
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Correspondence to Fatemeh Mohammadi .

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Editors and Affiliations

  1. Universität Kaiserslautern , Kaiserslautern, Germany

    Gert-Martin Greuel

  2. Zuse Institute Berlin , Berlin, Germany

    Thorsten Koch

  3. Johannes Kepler University Linz , Hagenberg, Austria

    Peter Paule

  4. University of Notre Dame , Notre Dame, Indiana, USA

    Andrew Sommese

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Mohammadi, F. (2016). Combinatorial and Geometric View of the System Reliability Theory. In: Greuel, GM., Koch, T., Paule, P., Sommese, A. (eds) Mathematical Software – ICMS 2016. ICMS 2016. Lecture Notes in Computer Science(), vol 9725. Springer, Cham. https://doi.org/10.1007/978-3-319-42432-3_19

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  • DOI: https://doi.org/10.1007/978-3-319-42432-3_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42431-6

  • Online ISBN: 978-3-319-42432-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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