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Conference on Algorithms and Discrete Applied Mathematics

CALDAM 2017: Algorithms and Discrete Applied Mathematics pp 294–307Cite as

The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract)

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10156))

Abstract

Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not “symmetric”, e.g., their graphs are not vertex transitive, not even regular.

We show that in the graph of the pedigree polytope, the quotient minimum degree over number of vertices tends to 1 as the number of cities tends to infinity.

D.O. Theis—Supported by the Estonian Research Council, ETAG (Eesti Teadusagentuur), through PUT Exploratory Grant #620, and by the European Regional Development Fund through the Estonian Center of Excellence in Computer Science, EXCS.

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Notes

  1. 1.

    We speak of vertices of the pedigree graph and nodes of the cycles, to limit confusion.

  2. 2.

    The reason why we use this notion of “infinite cycle” is pure convenience. It does not add complexity, but it makes many of statements and proofs less cumbersome. Indeed, instead of an infinite cycle, it is ok to just use a cycle whose length M is longer than all the lengths occuring in the particular argument. So instead of “let A be an infinite cycle, and consider \(A_k\), \(A_\ell \), \(A_n\)” you have to say “let M be a large enough integer, \(A_M\) a cycle of length M, and \(A_k\), \(A_\ell \), \(A_n\) sub-cycles of \(A_M\)”. All the little arguments (e.g., Fact 8 below) have to be done in the same way.

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Authors and Affiliations

  1. Institute of Computer Science, University of Tartu, Ülikooli 17, 51014, Tartu, Estonia

    Abdullah Makkeh, Mozhgan Pourmoradnasseri & Dirk Oliver Theis

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  1. Abdullah Makkeh
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  2. Mozhgan Pourmoradnasseri
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  3. Dirk Oliver Theis
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Correspondence to Dirk Oliver Theis .

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Editors and Affiliations

  1. University of Lethbridge, Lethbridge, Alberta, Canada

    Daya Gaur

  2. Indian Institute of Technology Madras, Chennai, India

    N.S. Narayanaswamy

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Makkeh, A., Pourmoradnasseri, M., Theis, D.O. (2017). The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract). In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_26

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  • DOI: https://doi.org/10.1007/978-3-319-53007-9_26

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