  The dominating set problem has many practical applications but is well-known
to be NP-hard. Therefore, there is a need for efficient approximation
algorithms, especially in applications such as ad hoc wireless networks. Most
distributed algorithms proposed in the literature assume that each node has
knowledge of the network structure. We propose a distributed approximation
algorithm that uses two rounds of communication, and where each node has only
local information, both in terms of network structure and dominating set
assignment. First, each node calculates a local centrality measure to determine
whether it is part of the dominating set $D$. The second round guarantees $D$
is a dominating set by adding any non-dominated nodes. We compare several
centrality measures and show that the Shapley value, introduced in game theory,
is theoretically motivated and performs well in practice on several synthetic
and real-world networks.
