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Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

Author

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  • Andrei V. Nikolaev

    (P.G. Demidov Yaroslavl State University)

  • Egor V. Klimov

    (P.G. Demidov Yaroslavl State University)

Abstract

A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. We consider the second Hamiltonian decomposition problem: for a 4-regular multigraph, find 2 edge-disjoint Hamiltonian cycles different from the given ones. This problem arises in polyhedral combinatorics as a sufficient condition for non-adjacency in the 1-skeleton of the traveling salesperson polytope. We introduce two integer linear programming models for the problem based on the classical Dantzig-Fulkerson-Johnson and Miller-Tucker-Zemlin formulations for the traveling salesperson problem. To enhance the performance on feasible problems, we supplement the algorithm with a variable neighborhood descent heuristic w.r.t. two neighborhood structures and a chain edge fixing procedure. Based on the computational experiments, the Dantzig-Fulkerson-Johnson formulation showed the best results on directed multigraphs, while on undirected multigraphs, the variable neighborhood descent heuristic was especially effective.

Suggested Citation

  • Andrei V. Nikolaev & Egor V. Klimov, 2024. "Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming," Journal of Combinatorial Optimization, Springer, vol. 47(5), pages 1-31, July.
    • Handle: RePEc:spr:jcomop:v:47:y:2024:i:5:d:10.1007_s10878-024-01184-0
    DOI: 10.1007/s10878-024-01184-0
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    References listed on IDEAS

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