Learning Automata-Based Algorithms for Finding Minimum Weakly Connected Dominating Set in Stochastic Graphs

A weakly connected dominating set (WCDS) of graph is a subset of so that the vertex set of the given subset and all vertices with at least one endpoint in the subset induce a connected sub‐graph of . Finding the WCDS is a new promising approach for clustering the wireless networks. The minimum WCDS (MWCDS) problem is known to be NP‐hard, and several approximation algorithms have been proposed for solving MWCDS in deterministic graphs. However, to the best of our knowledge no work has been done on finding the WCDS in stochastic graphs. In this paper, a definition of the MWCDS problem in a stochastic graph is first presented and then several learning automata‐based algorithms are proposed for solving the stochastic MWCDS problem where the probability distribution function of the weight associated with the graph vertices is unknown. Taking advantage of learning automata, the proposed algorithms significantly reduce the number of samples needs to be taken from the vertices of the stochastic graph. It is shown that by a proper choice of the parameters of the proposed algorithms, the probability of finding the MWCDS is as close to unity as possible. To evaluate the proposed algorithms, the number of samples taken from the graph is compared with that of the standard sampling method (SSM). Experimental results show the major superiority of the proposed algorithms over the SSM in terms of the sampling rate. Keyword Weakly connected dominating set, stochastic graph, learning automata, distributed learning automata.