Constant round distributed domination on graph classes with bounded expansion

Abstract

We show that the dominating set problem admits a constant factor approximation in a constant number of rounds in the LOCAL model of distributed computing on graph classes with bounded expansion. This generalizes a result of Czygrinow et al. for graphs with excluded topological minors.

Local planar domination revisited

“…Czygrinow et al [5] defined the notion of pseudo-covers, which provide a tool to carry out a fine grained analysis of vertices that can potentially belong to the sets A v used to dominate the red neighborhood N R pvq of a vertex v. This tool can in fact be applied to much more general graphs than planar graphs, namely, to all graphs that exclude some complete bipartite graph K t,t . A refined analysis for classes of bounded expansion was provided by Kublenz et al [10]. We provide an even finer analysis for planar graphs on which we base a second phase of our distributed algorithm.…”

“…By defining the notion of pseudo-covers, Czygrinow et al [5] provided a tool to carry out a fine grained analysis of the vertices that can potentially dominate the remaining neighborhoods. Using ideas of [10] and [16] we provide an even finer analysis for planar graphs on which we base the second phase of our distributed algorithm and compute a second partial dominating set.…”

A tight local algorithm for the minimum dominating set problem in outerplanar graphs

“…The current best known upper-bound is 52 [18], while the best lower-bound is 7 [10]. Substantial work has focused on generalizing the fact that some constant factor approximation is possible to more general classes of sparse graphs, like graphs that can be embedded on a given surface, or more recently graphs of bounded expansion [1,2,5,11]. Tight bounds currently seem out of reach in those more general contexts.…”