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 ∪ 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, SICOMP 1981. Recently, a 3/2-approximation was given for the unweighted case by Kortsarz and Nutov, TALG 2016, and recent breakthroughs by Adjiashvili, SODA 2017, and by Fiorini et al., 2017, give approximations better than 2 for bounded weights. 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 + √ 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(hMST + √ n log∗ n)-round 3-approximation algorithm for the weighted case, where hMST 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.
|Title of host publication||21st International Conference on Principles of Distributed Systems, OPODIS 2017|
|Editors||James Aspnes, Joao Leitao, Alysson Bessani, Pascal Felber|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Mar 2018|
|Event||21st International Conference on Principles of Distributed Systems, OPODIS 2017 - Lisboa, Portugal|
Duration: 18 Dec 2017 → 20 Dec 2017
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||21st International Conference on Principles of Distributed Systems, OPODIS 2017|
|Period||18/12/17 → 20/12/17|
Bibliographical noteFunding Information:
∗ Supported in part by the Israel Science Foundation (grant 1696/14). † The full version of the paper is available at https://arxiv.org/abs/1711.03359.
© 2017 Keren Censor-Hillel and Michal Dory.
- Approximation algorithms
- Connectivity augmentation
- Distributed network design
ASJC Scopus subject areas