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

  • Andrei NikolaevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Traveling salesperson problem Directed graph Pyramidal tour with step-backs Polytope 1-skeleton Vertex adjacency 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.P.G. Demidov Yaroslavl State UniversityYaroslavlRussia

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