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arXiv · 2201.03846

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

Andrei V. Nikolaev·Egor V. Klimov
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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 travelling 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 travelling salesperson problem. To enhance the performance on feasible problems, we supplement the algorithm with a variable neighbourhood descent heuristic w.r.t. two neighbourhood 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 neighbourhood descent heuristic was especially effective.

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Andrei V. Nikolaev, Egor V. Klimov. 2022-01-11. Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming. https://doi.org/10.1007/s10878-024-01184-0

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