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 Erd$\acute{o}$s R$\acute{e}$nyi (ER) random graph and regular random graph, but this work can be extended to any type of networks. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation algorithm does not converge when the inverse temperature exceeds a threshold on either an ER random network or regular random network. Second, the entropy density of replica symmetric theory has a transition point at a finite inverse temperature on a regular random graph when the node degree exceeds 4 and on an ER random graph when the node degree exceeds 6.6; there is no entropy transition point (or $\beta=\infty$) in other circumstances. Third, the results of the replica symmetry (RS) theory are in perfect agreement with those of belief propagation (BP) algorithm while the results of the belief propagation decimation (BPD) algorithm are better than those of the greedy heuristic algorithm.