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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.