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International Conference on Mathematical Optimization Theory and Operations Research

MOTOR 2019: Mathematical Optimization Theory and Operations Research pp 247–263Cite as

On Vertex Adjacencies in the Polytope of Pyramidal Tours with Step-Backs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11548))

Abstract

We consider the traveling salesperson problem in a directed graph. The pyramidal tours with step-backs are a special class of Hamiltonian tours for which the traveling salesperson problem is solved by dynamic programming in polynomial time. The polytope of pyramidal tours with step-backs \(\mathrm{{PSB}}(n)\) is defined as the convex hull of the characteristic vectors of all possible pyramidal tours with step-backs in a complete directed graph. The skeleton of \({\mathrm{{PSB}}} (n)\) is the graph whose vertex set is the vertex set of \({\mathrm{{PSB}}} (n)\) and the edge set is the set of geometric edges or one-dimensional faces of \({\mathrm{{PSB}}} (n)\). The main result of the paper is a necessary and sufficient condition for vertex adjacencies in the skeleton of the polytope \({\mathrm{{PSB}}} (n)\) that can be verified in polynomial time.

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Acknowledgments

The research is supported by the grant of the President of the Russian Federation MK-2620.2018.1 (agreement no. 075-015-2019-746).

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Authors and Affiliations

  1. P.G. Demidov Yaroslavl State University, Yaroslavl, Russia

    Andrei Nikolaev

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  1. Andrei Nikolaev
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Correspondence to Andrei Nikolaev .

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Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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Nikolaev, A. (2019). On Vertex Adjacencies in the Polytope of Pyramidal Tours with Step-Backs. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_18

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  • DOI: https://doi.org/10.1007/978-3-030-22629-9_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22628-2

  • Online ISBN: 978-3-030-22629-9

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