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 JaJa (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.
Highlights
The tree augmentation problem (TAP) is a central problem in network design
In TAP, the input is a 2-edge-connected1 graph G and a spanning tree T of G, and the goal is to augment T to be 2-edge-connected by adding to it a minimum size set of edges from G
The motivation for considering TAP is for the case that adding any new edge to the backbone incurs a cost, and if we are already given a subgraph with some connectivity guarantee we would naturally like to augment it with additional edges of minimum number or weight, rather than to compute a well-connected low-cost subgraph from scratch
Summary
In TAP, the input is a 2-edge-connected graph G and a spanning tree T of G, and the goal is to augment T to be 2-edge-connected by adding to it a minimum size (or a minimum weight) set of edges from G. The motivation for considering TAP is for the case that adding any new edge to the backbone incurs a cost, and if we are already given a subgraph with some connectivity guarantee we would naturally like to augment it with additional edges of minimum number or weight, rather than to compute a well-connected low-cost subgraph from scratch. In addition to fast approximations for TAP, our algorithms have the crucial implication of providing efficient algorithms for approximating the minimum 2-edge-connected spanning subgraph, as well as several related problems, such as verifying 2-edge-connectivity and augmenting the connectivity of any spanning connected subgraph to 2. We complement our study with proving lower bounds for distributed approximations of TAP
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