Distributed minimum dominating set approximations in restricted families of graphs

• Published in: Distributed computing Vol. 26; no. 2; pp. 119 - 137
• Main Authors: Lenzen, Christoph, Pignolet, Yvonne-Anne, Wattenhofer, Roger
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.04.2013; Springer; Springer Nature B.V
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Asymptotic properties; Computer Communication Networks; Computer Hardware; Computer networks; Computer Science; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Computer Systems Organization and Communication Networks; Exact sciences and technology; Forests; Graphs; Information retrieval. Graph; Planes; Software; Software Engineering/Programming and Operating Systems; Theoretical computing; Theory of Computation; Tradeoffs; Lower bound; Planar; Upper bound; Bounded arboricity; Unit disk; Dominating set; Euclidean theory; Planar graph; Tree structure; Distributed system; Approximation algorithm; Randomization; Spanning tree; Wireless network; Asymptotic approximation; Deterministic approach; Time complexity; Directed graph; Deterministic algorithms
• ISSN: 0178-2770; 1432-0452
• DOI: 10.1007/s00446-013-0186-z

A dominating set is a subset of the nodes of a graph such that all nodes are in the set or adjacent to a node in the set. A minimum dominating set approximation is a dominating set that is not much larger than a dominating set with the fewest possible number of nodes. This article summarizes the state-of-the-art with respect to finding minimum dominating set approximations in distributed systems, where each node locally executes a protocol on its own, communicating with its neighbors in order to achieve a solution with good global properties. Moreover, we present a number of recent results for specific families of graphs in detail. A unit disk graph is given by an embedding of the nodes in the Euclidean plane, where two nodes are joined by an edge exactly if they are in distance at most one. For this family of graphs, we prove an asymptotically tight lower bound on the trade-off between time complexity and approximation ratio of deterministic algorithms. Next, we consider graphs of small arboricity , whose edge sets can be decomposed into a small number of forests. We give two algorithms, a randomized one excelling in its approximation ratio and a uniform deterministic one which is faster and simpler. Finally, we show that in planar graphs , which can be drawn in the Euclidean plane without intersecting edges, a constant approximation factor can be ensured within a constant number of communication rounds.