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We describe a simple deterministic O(e-1 log A) round distributed algorithm for (2a + 1)(1 + e) approximation of minimum weighted dominating set on graphs with arboricity at most a. Here A denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized O(a(2)) approximation in O(log n) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic O(a log ?) approximation in O(log A) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic O(a) approximation in O(log(2)?) rounds (implicit in Bansal et al. in Inform Process Lett 122:21-24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized O(a) approximation in O(a log n) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized O(a log ?) round distributed algorithm that sharpens the approximation factor to a (1 + o(1)). If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve a - 1 - e approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017)., Distributed Computing, ISSN:0178-2770, ISSN:1432-0452

We consider the minimum dominating set problem on graphs with bounded arboricity. For graphs with arboricity a, we give an LP rounding algorithm that achieves an approximation factor of 3a and we complement this with a matching (up to constants) hardness of approximation result.

Given a graph G = (V, E), a subgraph G' = (V, H), H ⊆ E is a k-spanner of G if for any pair of vertices u, w ∈ V it satisfies dH(u, w) ≤ kdG(u, w). The basic k-spanner problem is to find a k-spanner of a given graph G with the smallest possible number of edges. This paper considers approximation algorithms for this and some related problems for k > 2, known to be Ω(2log1-µn)-inapproximable. The basic k-spanner problem over undirected graphs with k > 2 has been given a sublinear ratio approximation algorithm (with ratio roughly O(n2/(k+1))), but no such algorithms were known for other variants of the problem, including the directed and the client-server variants, as well as for the related k-DSS problem. We present the first approximation algorithms for these problems with sublinear approximation ratio. The second contribution of this paper is in characterizing some wide families of graphs on which the problems do admit a logarithmic and a polylogarithmic approximation ratios. These families are characterized as containing graphs that have optimal or "near-optimal" spanners with certain desirable properties, such as being a tree, having low arboricity or having low girth. All our results generalize to the directed and the client-server variants of the problems. As a simple corollary, we present an algorithm that given a graph G builds a subgraph with O(n) edges and stretch bounded by the tree-stretch of G, namely the minimum maximal stretch of a spanning tree for G. The analysis of our algorithms involves the novel notion of edge-dominating systems developed in the paper. The technique introduced in the paper reduces the studied algorithmic approximability questions on k-spanners to purely graph-theoretical questions concerning the existence of certain combinatorial objects in families of graphs.