Optimal Vertex Cover for the Small-World Hanoi Networks

Details Optimal Vertex Cover for the Small-World Hanoi Networks Optimal Vertex Cover for the Small-World Hanoi Networks

2011-03-30

arxiv.org/abs/1103.5940v2

Physical Review E 84, 011108 (2011)

Physics

Condensed Matter

Disordered Systems and Neural Networks

RevTex, 24 pages; many corrections in text and figures; final version; for related information, see http://www.physics.emory

Scientific paper

10.1103/PhysRevE.84.011108

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.

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