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Fast distributed approximation for TAP and 2-edge-connectivity

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Abstract

The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph G and a spanning tree T for it, and the goal is to augment T with a minimum set of edges Aug from G, such that \(T \cup Aug\) is 2-edge-connected. TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and JáJá (SIAM J Comput 10(2):270–283, 1981). Recently, a 3/2-approximation was given for unweighted TAP by Kortsarz and Nutov (ACM Trans Algorithms 12(2):23, 2016). Recent breakthroughs give an approximation of 1.458 for unweighted TAP (Grandoni et al. in: Proceedings of the 50th annual ACM SIGACT symposium on theory of computing (STOC 2018), 2018), and approximations better than 2 for bounded weights (Adjiashvili in: Proceedings of the twenty-eighth annual ACM-SIAM symposium on discrete algorithms (SODA), 2017; Fiorini et al. in: Proceedings of the twenty-ninth annual ACM-SIAM symposium on discrete algorithms (SODA 2018), New Orleans, LA, USA, 2018. https://doi.org/10.1137/1.9781611975031.53). In this paper, we provide the first fast distributed approximations for TAP. We present a distributed 2-approximation for weighted TAP which completes in O(h) rounds, where h is the height of T. When h is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in \(O(D+\sqrt{n}\log ^*{n})\) rounds, where n is the number of vertices and D is the diameter of G. Immediate consequences of our results are an O(D)-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an \(O(h_{MST}+\sqrt{n}\log ^{*}{n})\)-round 3-approximation algorithm for the weighted case, where \(h_{MST}\) is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2. Finally, we complement our study with proving lower bounds for distributed approximations of TAP.

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Notes

  1. A graph G is 2-edge-connected if it remains connected after the removal of any single edge.

  2. A directed spanning tree of G rooted at r, is a subgraph T of G such that the undirected version of T is a tree and T contains a directed path from r to any other vertex in V. A directed MST is a directed spanning tree of minimum weight.

  3. If a root and orientation are not given, we can find a root r and orient all the edges towards r in O(h) rounds using standard techniques.

  4. We assume that the MST is unique. Otherwise, \(h_{MST}\) is the height of the MST we construct.

  5. A verification algorithm with the same complexity can also be deduced from the edge-biconnectivity algorithm of Pritchard [33].

  6. We also show a construction with no parallel edges.

  7. Notice that at least one of the two parallel edges indeed has weight 0.

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  1. Department of Computer Science, Technion, Haifa, Israel

    Keren Censor-Hillel & Michal Dory

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  1. Keren Censor-Hillel
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  2. Michal Dory
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Correspondence to Michal Dory.

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A preliminary version of this paper appeared in OPODIS 2017.

Keren Censor-Hillel and Michal Dory: Supported in part by the Israel Science Foundation (Grant 1696/14). This project has received funding from the European Union’s Horizon 2020 Research And Innovation Program under grant agreement no.755839.

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Censor-Hillel, K., Dory, M. Fast distributed approximation for TAP and 2-edge-connectivity. Distrib. Comput. 33, 145–168 (2020). https://doi.org/10.1007/s00446-019-00353-3

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