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The minimal dominating Set (MDS) problem is a prototypical hard combinatorialoptimization problem. Two years ago we studied this problem by cavity method.Although we get the solution of a given graph, which gives very good estimationof minimal dominating size, but we don't know whether we get the ground statesolution 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 that1) The MDS problem solution space has both condensation transition and clustertransition on regular Random (RR) graph and we prove this by simulatedannealing dynamical process. 2) We developed zero temperature SurveyPropagation (SP) algorithm on ER graph to estimate the ground state energy andto get Survey Propagation Decimation (SPD) algorithm with good results same asBPD algorithm.

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