Brief announcement: deterministic dominating set construction in networks with bounded degree

This paper considers the problem of calculating dominating sets in bounded degree networks. In these networks, the maximal degree of any node is bounded by δ, which is usually significantly smaller than n, the total number of nodes in the system. Such networks arise in various settings of wireless and peer-to-peer communication. A trivial approach of choosing all nodes into the dominating set yields an algorithm with an approximation ratio of δ + 1. We show that any deterministic algorithm with a non-trivial approximation ratio requires Ω(log* n) rounds, meaning effectively that no local o(δ)-approximation deterministic algorithm may ever exist. On the positive side, we show two deterministic algorithms that achieve log δ and 2 log δ-approximation in O(δ3 + log* n) and O(δ2 logδ + log* n) time, respectively. These algorithms rely on coloring rather than node IDs to break symmetry.