Approximating k-Edge-Connected Spanning Subgraphs via a Near-Linear Time LP Solver

In the k-edge-connected spanning subgraph (kECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to k link failures: Given an n-node m-edge graph with a cost function on the edges, our goal is to compute a minimum-cost k-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the “uniform case” of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a special case of 2ECSS. The fastest 2-approximation algorithm is however rather slow, taking O(mnk) time [Khuller, Vishkin, STOC'92]. A faster time complexity of O(n²) can be obtained, but with a higher approximation guarantee of (2k − 1) [Gabow, Goemans, Williamson, IPCO'93]. Our main contribution is an algorithm that (1 + ε)-approximates the optimal fractional solution in Õ(m/ε²) time (independent of k), which can be turned into a (2 + ε) approximation algorithm that runs in time (Equation presented) for (integral) kECSS; this improves the running time of the aforementioned results while keeping the approximation ratio arbitrarily close to a factor of two.