M. Weigt et Ak. Hartmann, Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture - art. no. 056127, PHYS REV E, 6305(5), 2001, pp. 6127
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 1, 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 a
nd entropy of the minimal vertex covers, and the structure of the unfrozen
component which is found to percolate at a connectivity c similar or equal
to 1.43. The replica-symmetric solution breaks down at c = e similar or equ
al to 2.72. We give a simple one-step replica-symmetry-broken solution, and
discuss the problems in the interpretation and generalization of this solu
tion.