# Distributed Distance-r Dominating Set on Sparse High-Girth Graphs

The dominating set problem and its generalization, the distance-r dominating set problem, are among the well-studied problems in the sequential settings. In distributed models of computation, unlike for domination, not much is known about distance-r domination. This is actually the case for other important closely-related covering problem, namely, the distance-r independent set problem. By result of Kuhn et al. we know the distributed domination problem is hard on high girth graphs; we study the problem on a slightly restricted subclass of these graphs: graphs of bounded expansion with high girth, i.e. their girth should be at least 4r + 3. We show that in such graphs, for every constant r, a simple greedy CONGEST algorithm provides a constant-factor approximation of the minimum distance-r dominating set problem, in a constant number of rounds. More precisely, our constants are dependent to r, not to the size of the graph. This is the first algorithm that shows there are non-trivial constant factor approximations in constant number of rounds for any distance r-covering problem in distributed settings. To show the dependency on r is inevitable, we provide an unconditional lower bound showing the same problem is hard already on rings. We also show that our analysis of the algorithm is relatively tight, that is any significant improvement to the approximation factor requires new algorithmic ideas.

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• 6 publications
• 2 publications
• ### Constant round distributed domination on graph classes with bounded expansion

We show that the dominating set problem admits a constant factor approxi...
12/04/2020 ∙ by Simeon Kublenz, et al. ∙ 0

• ### On distance r-dominating and 2r-independent sets in sparse graphs

Dvorak (2013) gave a bound on the minimum size of a distance r dominatin...
10/27/2017 ∙ by Zdeněk Dvořák, et al. ∙ 0

• ### Distributed Reconfiguration of Maximal Independent Sets

Consider the following problem: given a graph and two maximal independen...
10/04/2018 ∙ by Keren Censor-Hillel, et al. ∙ 0

• ### Greedy domination on biclique-free graphs

The greedy algorithm for approximating dominating sets is a simple metho...
06/07/2018 ∙ by Sebastian Siebertz, et al. ∙ 0

• ### Deterministic CONGEST Algorithm for MDS on Bounded Arboricity Graphs

We provide a deterministic CONGEST algorithm to constant factor approxim...
02/16/2021 ∙ by Saeed Akhoondian Amiri, et al. ∙ 0

• ### Experimental evaluation of kernelization algorithms to Dominating Set

The theoretical notions of graph classes with bounded expansion and that...
11/19/2018 ∙ by Wojciech Nadara, et al. ∙ 0