Approximation algorithms for flexible graph connectivity

We present approximation algorithms for several network design problems in the model of flexible graph connectivity (Adjiashvili et al., in: IPCO, pp 13-26, 2020, Math Program 1-33, 2021). Let k >== 1, p >= 1 and q >= 0 be integers. In an instance of the ( p, q)-Flexible Graph Connectivity problem, denoted ( p, q)-FGC, we have an undirected connected graph G = (V, E), a partition of E into a set of safe edges l and a set of unsafe edges U, and nonnegative costs c : E -> R(>=)0 on the edges. A subset F subset of. E of edges is feasible for the ( p, q)-FGC problem if for any set F 'subset of. U with | F '| <= q, the subgraph (V, F\ F ') is p-edge connected. The algorithmic goal is to find a feasible solution F that minimizes c( F) = Sigma F-e is an element of(ce.) We present a simple 2-approximation algorithm for the (1, 1)-FGC problem via a reduction to the minimum-cost rooted 2-arborescence problem. This improves on the 2.527-approximation algorithm of Adjiashvili et al. Our 2-approximation algorithm for the (1, 1)-FGC problem extends to a (k + 1)-approximation algorithm for the (1, k)-FGC problem. We present a 4-approximation algorithm for the (k, 1)-FGC problem, and an O(q log | V|)-approximation algorithm for the ( p, q)-FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted (1, 1)-FGC problem by presenting a 16/11-approximation algorithm. The ( p, q)-FGC problem is related to the well-known Capacitated k-Connected Subgraph problem (denoted Cap-k-ECSS) that arises in the area of Capacitated Network Design. We give a min(k, 2u(max))-approximation algorithm for the Cap-k-ECSS problem, where u(max) denotes the maximum capacity of an edge.