Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

Abstract

Since in general it is NP-hard to solve the minimum dominating set problem even approximatively, a lot of work has been dedicated to central and distributed approximation algorithms on restricted graph classes. In this paper, we compromise between generality and efficiency by considering the problem on graphs of small arboricity a. These family includes, but is not limited to, graphs excluding fixed minors, such as planar graphs, graphs of (locally) bounded treewidth, or bounded genus. We give two viable distributed algorithms. Our first algorithm employs a forest decomposition, achieving a factor \({\mathcal{O}}(a^2)\) approximation in randomized time \({\mathcal{O}}(\log n)\). This algorithm can be transformed into a deterministic central routine computing a linear-time constant approximation on a graph of bounded arboricity, without a priori knowledge on a. The second algorithm exhibits an approximation ratio of \({\mathcal{O}}(a\log \Delta)\), where Δ is the maximum degree, but in turn is uniform and deterministic, and terminates after \({\mathcal{O}}(\log \Delta)\) rounds. A simple modification offers a trade-off between running time and approximation ratio, that is, for any parameter α ≥ 2, we can obtain an \({\mathcal{O}}(a \alpha \log_{\alpha} \Delta)\)-approximation within \({\mathcal{O}}(\log_{\alpha} \Delta)\) rounds.