Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning Subgraph

The minimum-weight 2-edge-connected spanning subgraph (2-ECSS) problem is a natural generalization of the well-studied minimum-weight spanning tree (MST) problem, and it has received considerable attention in the area of network design. The latter problem asks for a minimum-weight subgraph with an edge connectivity of 1 between each pair of vertices while the former strengthens this edge-connectivity requirement to 2. Despite this resemblance, the 2-ECSS problem is considerably more complex than MST. While MST admits a linear-time centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized approximation algorithm for it (that runs in polynomial time) gives a 2-approximation. In this paper, we give a deterministic distributed algorithm with round complexity of Õ (D + √n) that computes a (9 + ε)-approximation of 2-ECSS, for any constant ε > 0. Up to logarithmic factors, this complexity matches the Ø (D + √ n) lower bound that can be derived from the technique of Das Sarma et al. [STOC'11], as shown by Censor-Hillel and Dory [OPODIS'17]. Our result is the first distributed constant approximation for 2-ECSS in the nearly optimal time and it improves on a recent randomized algorithm of Dory [PODC'18], which achieved an O(łog n)-approximation in Õ (D+√ ) rounds. We also present an alternative algorithm for O(log n)-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network---following a framework introduced by Ghaffari and Haeupler [SODA'16]. This algorithm has round complexity Ö (D+√n) in worst-case networks but it provably runs much faster in many well-behaved graph families of interest. For instance, it runs in Õ (D) time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.