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

TL;DR

This paper presents a distributed algorithm that efficiently approximates the minimum distance-$r$ dominating set in high-girth, bounded expansion graphs, demonstrating the problem's complexity and providing the first constant-factor approximation in constant rounds.

Contribution

It introduces the first constant-factor approximation algorithm for the distributed distance-$r$ dominating set problem on certain sparse graphs, with tight analysis and lower bounds.

Findings

Algorithm achieves constant-factor approximation in constant rounds.

Lower bounds show the problem's inherent difficulty on rings.

Analysis indicates significant improvements require new methods.

Abstract

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.