Distributed distance-r covering problems on sparse high-girth graphs

Distributed distance domination in graphs with no $K_{2,t}$-minor

“…We will work in the Local model of computations and assume throught the paper that k ≥ 2. Although the first algorithm is identical to the algorithm from [3] which works in the Congest model, the algorithm of Amiri et al exploits the fact that graphs are locally trees to allow for a Congest model implementation. Since the graphs considered in this paper can have many short cycles, the algorithm works only in the Local model.…”

“…The main motivation for our work comes from the recent paper by Amiri and Wiederhake [3] who managed to provide a first constant approximation algorithm in a constant number of rounds for distance-k domination in graphs of bounded expansion of high girth (i.e. graphs that are sparse and are trees locally).…”

“…graphs that are sparse and are trees locally). In fact, we will use the very same procedure from [3], but give a different argument in the first part of the paper as we will examine a different class of graphs. Note that the girth assumption in [3] was related to a previous work on lower bounds that were established for graphs of high girths and is absolutely critical to their analysis.…”

“…In fact, we will use the very same procedure from [3], but give a different argument in the first part of the paper as we will examine a different class of graphs. Note that the girth assumption in [3] was related to a previous work on lower bounds that were established for graphs of high girths and is absolutely critical to their analysis. It is this assumption that we will get rid of in the current paper (at the expense of dealing with graphs with no K 2,tminor rather than a much more general class of graphs of bounded expansion).…”

Deterministic CONGEST Algorithm for MDS on Bounded Arboricity Graphs

“…Amiri et al [2], additionally provided a constant factor approximation on bounded expansion graphs in logarithmic rounds for a generalized version of the problem: the distance-r MDS. The latter recently has been improved in two directions: Kublenz et al [17], reduced the number of rounds to a constant but only for the standard MDS-problem, Amiri and Wiederhake [6] showed that for high girth graphs the approximation algorithm for distance r-MDS can be obtained by O(r) rounds. The minimum girth requirement showed to be useful in a recent work of Alipour and Jafari [1], where they showed that in C 4 -free planar graphs there is a better approximation guarantee for the MDS problem.…”