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 thewell-studied minimumweight 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 HO(D+√n) that computes a (9+ϵ)-approximation of 2-ECSS, for any constant ϵ > 0. Up to logarithmic factors, this complexity matches the Ω(D + √n) lower bound that can be derived from the technique of Das Sarma et al. [STOC f11], as shown by Censor-Hillel and Dory [OPODIS f17]. 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 f18], which achieved an O(logn)-approximation in HO(D +√n) rounds. We also present an alternative algorithm forO(logn)-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network.following a framework introduced by Ghaffari and Haeupler [SODA f16]. This algorithm has round complexity HO(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 HO(D) time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.