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Simulated Annealing Approach to Verify Vertex Adjacencies in the Traveling Salesperson Polytope

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Mathematical Optimization Theory and Operations Research (MOTOR 2019)

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

Abstract

We consider 1-skeletons of the symmetric and asymmetric traveling salesperson polytopes whose vertices are all possible Hamiltonian tours in the complete directed or undirected graph, and the edges are geometric edges or one-dimensional faces of the polytope. It is known that the question whether two vertices of the symmetric or asymmetric traveling salesperson polytopes are nonadjacent is NP-complete. A sufficient condition for nonadjacency can be formulated as a combinatorial problem: if from the edges of two Hamiltonian tours we can construct two complementary Hamiltonian tours, then the corresponding vertices of the traveling salesperson polytope are not adjacent. We consider a heuristic simulated annealing approach to solve this problem. It is based on finding a vertex-disjoint cycle cover and a perfect matching. The algorithm has a one-sided error: the answer “not adjacent” is always correct, and was tested on random and pyramidal Hamiltonian tours.

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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

    Anna Kozlova & Andrei Nikolaev

Authors
  1. Anna Kozlova
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  2. 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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Kozlova, A., Nikolaev, A. (2019). Simulated Annealing Approach to Verify Vertex Adjacencies in the Traveling Salesperson Polytope. 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_26

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  • DOI: https://doi.org/10.1007/978-3-030-22629-9_26

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  • Online ISBN: 978-3-030-22629-9

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