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 $$T \cup Aug$$ T ∪ A u g 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 $$O(D+\sqrt{n}\log ^*{n})$$ O ( D + 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 $$O(h_{MST}+\sqrt{n}\log ^{*}{n})$$ O ( h MST + n log ∗ n ) -round 3-approximation algorithm for the weighted case, where $$h_{MST}$$ 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.

中文翻译：

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TAP 和 2-edge-connectivity 的快速分布式近似

树增强问题（TAP）是一个基本的网络设计问题，其中输入是一个图 G 和一个生成树 T，目标是用 G 的最小边 Aug 来增强 T，使得 $ $T \cup Aug$$ T ∪ ug 是 2 边连通的。TAP 在顺序设置中得到了广泛的研究。加权案例中最著名的近似比率 2 可以追溯到 Frederickson 和 JáJá 的工作（SIAM J Comput 10(2):270–283, 1981）。最近，Kortsarz 和 Nutov 对未加权的 TAP 给出了 3/2 近似值（ACM Trans Algorithms 12(2):23, 2016）。最近的突破为未加权的 TAP 提供了 1.458 的近似值（Grandoni 等人在：第 50 届年度 ACM SIGACT 计算理论研讨会论文集（STOC 2018），2018 年），并且对于有界权重的近似值优于 2（Adjiashvili 在：2017年第28届ACM-SIAM离散算法研讨会（SODA）论文集；菲奥里尼等人。在：第 29 届 ACM-SIAM 年度离散算法研讨会论文集 (SODA 2018)，美国路易斯安那州新奥尔良，2018 年。https://doi.org/10.1137/1.9781611975031.53）。在本文中，我们为 TAP 提供了第一个快速分布式近似。我们提出了加权 TAP 的分布式 2-近似，它在 O ( h ) 轮中完成，其中 h 是 T 的高度。当 h 很大时，我们为未加权的情况展示了一个更快的 4-近似算法，在 $$O(D+\sqrt{n}\log ^*{n})$$O ( D + n log ∗ n ) 中完成轮数，其中 n 是顶点数， D 是 G 的直径。我们的结果的直接结果是一个 O ( D )-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, 这显着改善了先前近似算法的运行时间，并且 $$O(h_{MST}+\sqrt{n}\log ^{*}{n})$$ O ( h MST + n log ∗ n ) -round 3-approximation algorithm for the weighted case，其中 $$h_{MST}$$ h MST 是图的 MST 的高度。其他应用是验证 2 边连通性和将任何连接的生成子图的连通性增加到 2 的算法。最后，我们通过证明 TAP 的分布式近似的下界来补充我们的研究。