Approximation Algorithms for Flexible Graph Connectivity
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We present approximation algorithms for several network design problems in the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and Mühlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021), and IPCO 2020: pp. 13-26). 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 S and a set of unsafe edges U, and nonnegative costs c: E→ on the edges. A subset F ⊆ E of edges is feasible for the (p,q)-FGC problem if for any subset F' of unsafe edges 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) = ∑_e ∈ F c_e. 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 (p,1)-FGC problem, and an O(qlog|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,2 u_max)-approximation algorithm for the Cap-k-ECSS problem, where u_max denotes the maximum capacity of an edge.
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