Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture
Abstract
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 1.43. The replica-symmetric solution breaks down at c=e 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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