Eur. Phys. J. B 32, 265-270 (2003)
https://doi.org/10.1140/epjb/e2003-00096-4

Vertex cover problem studied by cavity method: Analytics and population dynamics

Max-Planck-Institute of Colloids and Interfaces, 14424 Potsdam, Germany

Corresponding author: ^a zhou@mpikg-golm.mpg.de

Received: 11 November 2002
Published online: 1 April 2003

Abstract

We study the vertex cover problem on finite connectivity random graphs by zero-temperature cavity method. The minimum vertex cover corresponds to the ground state(s) of a proposed Ising spin model. When the connectivity c > e=2.718282, there is no state for this system as the reweighting parameter y, which takes a similar role as the inverse temperature β in conventional statistical physics, approaches infinity; consequently the ground state energy is obtained at a finite value of y when the free energy function attains its maximum value. The minimum vertex cover size at given c is estimated using population dynamics and compared with known rigorous bounds and numerical results. The backbone size is also calculated.