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

TL;DR

This paper introduces ILP models and heuristics to find a second Hamiltonian decomposition in 4-regular multigraphs, with applications in polyhedral combinatorics and TSP polytope analysis.

Contribution

It presents two ILP formulations and heuristic enhancements for the second Hamiltonian decomposition problem in 4-regular multigraphs, advancing computational methods in this area.

Findings

Dantzig-Fulkerson-Johnson ILP performs best on directed multigraphs.

VND heuristic is highly effective on undirected multigraphs.

The methods facilitate analysis of the TSP polytope's structure.

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.