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 J{a}J{a}, SICOMP 1981. Recently, a 3/2-approximation was given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018], and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017; Fiorini et al., SODA 2018]. 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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