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A Hamiltonian decomposition of a regular graph is a partition of its edge setinto Hamiltonian cycles. The problem of finding edge-disjoint Hamiltoniancycles in a given regular graph has many applications in combinatorialoptimization and operations research. Our motivation for this problem comesfrom the field of polyhedral combinatorics, as a sufficient condition forvertex nonadjacency in the 1-skeleton of the traveling salesperson polytope canbe formulated as the Hamiltonian decomposition problem in a 4-regularmultigraph with one forbidden decomposition. In our approach, the algorithmstarts by solving the relaxed 2-matching problem, then iteratively generatessubtour elimination constraints for all subtours in the solution and solves thecorresponding ILP-model to optimality. The procedure is enhanced by the localsearch heuristic based on chain edge fixing and cycle merging operations. Inthe computational experiments, the iterative ILP algorithm showed comparableresults with the previously known heuristics on undirected multigraphs andsignificantly better performance on directed multigraphs.

An iterative ILP approach for constructing a Hamiltonian decomposition of a regular multigraph