Distributed node-weighted connected dominating set problems

The Minimum Connected Dominating Set (MCDS) problem is to find a subset of vertices in a given graph G such that the set is connected and any vertex of G is either in the set or adjacent to a node in the set. This problem is shown to be NP-Hard and the best polynomial time approximation ratio is O(log n) where n is the number of vertices. The MCDS problem and its derivations are of interest in many network applications such as finding a minimum size virtual backbone for routing and broadcasting in ad-hoc networks. In this paper, we consider node-weighted CDS problem where positive real valued weights are assigned to the vertices and the objective is to find a CDS with the minimum weight. We propose the first distributed algorithm for the problem and demonstrate its optimal O(log n) approximation ratio. We then consider the case where the node weights are random variables with unknown distributions and develop a distributed learning algorithm based on the multi-armed bandit theory. We show that the distributed learning algorithm offers the optimal logarithmic regret order with respect to the time horizon length.