Optimal Vertex Cover for the Small-World Hanoi Networks

TL;DR

This paper investigates the vertex-cover problem on Hanoi networks HN3 and HN5 using renormalization group analysis and Monte Carlo simulations, revealing complex solution structures and dominant solution landscapes in the thermodynamic limit.

Contribution

It provides an exact renormalization group framework and detailed Monte Carlo analysis to understand the structure of optimal vertex covers on Hanoi networks.

Findings

Fixed point analysis reveals complex solution space structure.

Optimal coverages are not related by simple symmetry.

In the thermodynamic limit, a dominant set of similar solutions emerges.

Abstract

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 non-trivial 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.