Minimal vertex covers on finiteconnectivity random graphs: A hardsphere latticegas picture
Abstract
The minimal vertexcover (or maximal independentset) 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 replicasymmetric phase, including the size and entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at a connectivity c~=1.43. The replicasymmetric solution breaks down at c=e~=2.72. We give a simple onestep replicasymmetrybroken solution, and discuss the problems in the interpretation and generalization of this solution.
 Publication:

Physical Review E
 Pub Date:
 May 2001
 DOI:
 10.1103/PhysRevE.63.056127
 arXiv:
 arXiv:condmat/0011446
 Bibcode:
 2001PhRvE..63e6127W
 Keywords:

 89.20.Ff;
 05.20.y;
 05.50.+q;
 02.60.Pn;
 Computer science and technology;
 Classical statistical mechanics;
 Lattice theory and statistics;
 Numerical optimization;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics
 EPrint:
 32 pages, 9 eps figures, to app. in PRE (01 May 2001)