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Distributed minimum dominating set approximations in restricted families of graphs

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Abstract

A dominating set is a subset of the nodes of a graph such that all nodes are in the set or adjacent to a node in the set. A minimum dominating set approximation is a dominating set that is not much larger than a dominating set with the fewest possible number of nodes. This article summarizes the state-of-the-art with respect to finding minimum dominating set approximations in distributed systems, where each node locally executes a protocol on its own, communicating with its neighbors in order to achieve a solution with good global properties. Moreover, we present a number of recent results for specific families of graphs in detail. A unit disk graph is given by an embedding of the nodes in the Euclidean plane, where two nodes are joined by an edge exactly if they are in distance at most one. For this family of graphs, we prove an asymptotically tight lower bound on the trade-off between time complexity and approximation ratio of deterministic algorithms. Next, we consider graphs of small arboricity, whose edge sets can be decomposed into a small number of forests. We give two algorithms, a randomized one excelling in its approximation ratio and a uniform deterministic one which is faster and simpler. Finally, we show that in planar graphs, which can be drawn in the Euclidean plane without intersecting edges, a constant approximation factor can be ensured within a constant number of communication rounds.

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Notes

  1. For central or parallel algorithms, one is typically satisfied with a guarantee that holds in expectation. In the distributed setting, this approach has two severe shortcomings. Firstly, one may not want to run multiple instances of the algorithm and take the best result by counting the number of nodes in the obtained MDS: while this will boost the probability of a good approximation ratio, it also incurs a large overhead in the running time if the graph has a large diameter. Secondly, the running times of distributed algorithms are typically small (all presented algorithms run in \(\mathcal O (\log n)\) time), hence obtaining strong probability bounds at a comparably low running time can be particularly challenging.

  2. Exceptions are algorithms where nodes learn about the entire neighborhood up to a certain distance and then solve a hard problem on this neighborhood. Note, however, that this approach is also in conflict with the goal of few, small messages.

  3. This result was claimed previously, in [28] by us and independently in [10] by others. Sadly, our algorithm was wrong and the proof from [10] is incomplete. In this article, we present corrected versions of our algorithm and proof.

  4. Note that the original paper [28] contained an error and the stated algorithm does not compute a constant MDS approximation. Moreover, the proof of the algorithm from [10] is incomplete.

  5. Trivially, one can try all combinations of six nodes, but planarity permits more efficient solutions.

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Acknowledgments

We would like to thank Topi Musto, Jukka Suomela, and the anonymous reviewers who helped in improving this article and its predecessors in many ways. This work has been partly supported by the Swiss National Fund (SNF) and the Society of Swiss Friends of the Weizmann Institute.

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Authors and Affiliations

  1. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Christoph Lenzen

  2. ABB Corporate Research, Dättwil-Baden, Switzerland

    Yvonne-Anne Pignolet

  3. Computer Engineering and Networks Laboratory (TIK), ETH Zurich, Switzerland

    Roger Wattenhofer

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  1. Christoph Lenzen
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  2. Yvonne-Anne Pignolet
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  3. Roger Wattenhofer
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Correspondence to Christoph Lenzen.

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Lenzen, C., Pignolet, YA. & Wattenhofer, R. Distributed minimum dominating set approximations in restricted families of graphs. Distrib. Comput. 26, 119–137 (2013). https://doi.org/10.1007/s00446-013-0186-z

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