Optimal vertex cover for the small-world Hanoi networks

The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with an exact renormalization group and parallel-tempering Monte Carlo simulations. The grand canonical partition function of the equivalent hard-core repulsive lattice-gas problem is recast first as an Ising-like canonical partition function, which allows for a closed set of renormalization-group equations. The flow of these equations is analyzed for the limit of infinite chemical potential, at which the vertex-cover problem is attained. The relevant fixed point and its neighborhood are analyzed and nontrivial results are obtained both for the coverage as well as for the ground-state entropy density, which indicates the complex structure of the solution space. Using special hierarchy-dependent operators in the renormalization group and Monte Carlo simulations, structural details of optimal configurations are revealed. These studies indicate that the optimal coverages (or packings) are not related by a simple symmetry. Using a clustering analysis of the solutions obtained in the Monte Carlo simulations, a complex solution space structure is revealed for each system size. Nevertheless, in the thermodynamic limit, the solution landscape is dominated by one huge set of very similar solutions.