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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+epsilon). Let Delta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log{Delta} / log log Delta) rounds, for constants epsilon in (0,1] and f in N^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms.
For constant values of f and epsilon, our algorithm improves over the (f+epsilon)-approximation algorithm of [Fabian Kuhn et al., 2006] whose running time is O(log Delta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(f log n) rounds, improving over the classical result of [Samir Khuller et al., 1994] that achieves a running time of O(f log^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [Koufogiannakis and Young, 2011].
We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f+epsilon)-approximate integral solution in O((1+f/log n)* ((log Delta)/(log log Delta) + (f * log M)^{1.01}* log epsilon^{-1}* (log Delta)^{0.01})) rounds, where f bounds the number of variables in a constraint, Delta bounds the number of constraints a variable appears in, and M=max {1, ceil[1/a_{min}]}, where a_{min} is the smallest normalized constraint coefficient. This improves over the results of [Fabian Kuhn et al., 2006] for the integral case, which combined with rounding achieves the same guarantees in O(epsilon^{-4}* f^4 * log f * log(M * Delta)) rounds.

Ran Ben-Basat, Guy Even, Ken-ichi Kawarabayashi, and Gregory Schwartzman. Optimal Distributed Covering Algorithms. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{benbasat_et_al:LIPIcs.DISC.2019.5, author = {Ben-Basat, Ran and Even, Guy and Kawarabayashi, Ken-ichi and Schwartzman, Gregory}, title = {{Optimal Distributed Covering Algorithms}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {5:1--5:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.5}, URN = {urn:nbn:de:0030-drops-113129}, doi = {10.4230/LIPIcs.DISC.2019.5}, annote = {Keywords: Distributed Algorithms, Approximation Algorithms, Vertex Cover, Set Cover} }

Document

**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds.
Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2.

Ran Ben-Basat, Ken-ichi Kawarabayashi, and Gregory Schwartzman. Parameterized Distributed Algorithms. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{benbasat_et_al:LIPIcs.DISC.2019.6, author = {Ben-Basat, Ran and Kawarabayashi, Ken-ichi and Schwartzman, Gregory}, title = {{Parameterized Distributed Algorithms}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {6:1--6:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.6}, URN = {urn:nbn:de:0030-drops-113135}, doi = {10.4230/LIPIcs.DISC.2019.6}, annote = {Keywords: Distributed Algorithms, Approximation Algorithms, Parameterized Algorithms} }

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