Statistical mechanics of the directed 2-distance minimal dominating set problem

The directed L-distance minimal dominating set(MDS) problem has wide practical applications in the fields of computer science and communication networks. Here, we study this problem from the perspective of purely theoretical interest. We only give results for an Erdós Rényi(ER)random graph and regular random(RR) graph, but this work can be extended to any type of network. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation(BP) algorithm does not converge when the inverse temperatureβ exceeds a threshold on either an ER random network or RR network. Second, the entropy density of replica symmetric theory has a transition point at a finite β on a regular random graph when the arc density exceeds 2 and on an ER random graph when the arc density exceeds3.3;there is no entropy transition point(or β = ■) in other circumstances. Third, the results of the replica symmetry(RS) theory are in agreement with those of BP algorithm while the results of the BP decimation algorithm are better than those of the greedy heuristic algorithm.

Directed Dominating Set Problem Studied by Cavity Method:Warning Propagation and Population Dynamics

The minimal dominating set for a digraph(directed graph) is a prototypical hard combinatorial optimization problem. In a previous paper, we studied this problem using the cavity method. Although we found a solution for a given graph that gives very good estimate of the minimal dominating size, we further developed the one step replica symmetry breaking theory to determine the ground state energy of the undirected minimal dominating set problem. The solution space for the undirected minimal dominating set problem exhibits both condensation transition and cluster transition on regular random graphs. We also developed the zero temperature survey propagation algorithm on undirected Erds-Rnyi graphs to find the ground state energy. In this paper we continue to develope the one step replica symmetry breaking theory to find the ground state energy for the directed minimal dominating set problem. We find the following.(i) The warning propagation equation can not converge when the connectivity is greater than the core percolation threshold value of 3.704. Positive edges have two types warning, but the negative edges have one.(ii) We determine the ground state energy and the transition point of the Erd?os-R′enyi random graph.(iii) The survey propagation decimation algorithm has good results comparable with the belief propagation decimation algorithm.