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

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

Keywords

Traveling salesperson problem Hamiltonian tour Traveling salesperson polytope 1-skeleton Vertex adjacency Simulated annealing Vertex-disjoint cycle cover Perfect matching 

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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