Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture

Details Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture

2000-11-27

arxiv.org/abs/cond-mat/0011446v2

Phys. Rev. E 63, 056127 (2001)

Physics

Condensed Matter

Disordered Systems and Neural Networks

32 pages, 9 eps figures, to app. in PRE (01 May 2001)

Scientific paper

10.1103/PhysRevE.63.056127

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity $c\simeq 1.43$. The replica-symmetric solution breaks down at $c=e\simeq 2.72$. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.

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