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Approximating (p,2) flexible graph connectivity via the primal-dual method

by   Ishan Bansal, et al.
University of Waterloo
cornell university

We consider the Flexible Graph Connectivity model (denoted FGC) introduced by Adjiashvili, Hommelsheim and Mühlenthaler (IPCO 2020, Mathematical Programming 2021), and its generalization, (p,q)-FGC, where p ≥ 1 and q ≥ 0 are integers, introduced by Boyd et al. (FSTTCS 2021). In the (p,q)-FGC model, we have an undirected connected graph G=(V,E), non-negative costs c on the edges, and a partition (𝒮, 𝒰) of E into a set of safe edges 𝒮 and a set of unsafe edges 𝒰. A subset F ⊆ E of edges is called feasible if for any set F'⊆𝒰 with |F'| ≤ q, the subgraph (V, F ∖ F') is p-edge connected. The goal is to find a feasible edge-set of minimum cost. For the special case of (p,q)-FGC when q = 2, we give an O(1) approximation algorithm, thus improving on the logarithmic approximation ratio of Boyd et al. (FSTTCS 2021). Our algorithm is based on the primal-dual method for covering an uncrossable family, due to Williamson et al. (Combinatorica 1995). We conclude by studying weakly uncrossable families, which are a generalization of the well-known notion of an uncrossable family.


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