Minimal Dominating Set problem studied by simulated annealing and cavity method: Analytics and population dynamics

Abstract

The minimal dominating Set (MDS) problem is a prototypical hard combinatorial optimization problem. Two years ago we studied this problem by cavity method. Although we get the solution of a given graph, which gives very good estimation of minimal dominating size, but we don't know whether we get the ground state solution and how many solutions exist in the ground state. For this purpose, last year we continue to develop the one step replica symmetry breaking (RSB) theory to find the ground state energy of the MDS problem. Finally we find that 1) The MDS problem solution space has both condensation transition and cluster transition on regular Random (RR) graph and we prove this by simulated annealing dynamical process. 2) We developed zero temperature Survey Propagation (SP) algorithm on ER graph to estimate the ground state energy and to get Survey Propagation Decimation (SPD) algorithm with good results same as BPD algorithm.

16pages,6figures,2tables

References in corpus (7)

• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• Clusters of solutions and replica symmetry breaking in random k-satisfiability
• Network Observability Transitions
• Distance-d covering problems in scale-free networks with degree correlations
• Statistical Mechanics of the Minimum Dominating Set Problem
• Long-range frustration in T=0 first-step replica-symmetry-broken solutions of finite-connectivity spin glasses
• Theoretical analysis of optimization problems - Some properties of random k-SAT and k-XORSAT