Extensions of the $(p,q)$-Flexible-Graph-Connectivity model

We present approximation algorithms for network design problems in some models related to the $(p,q)$-FGC model. Adjiashvili, Hommelsheim and Mühlenthaler introduced the model of Flexible Graph Connectivity that we denote by FGC. Boyd, Cheriyan, Haddadan and Ibrahimpur introduced a generalization of FGC. Let $p\geq 1$ and $q\geq 0$ 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 and a set of unsafe edges, and nonnegative costs $c\in\mathbb{R}_{\geq0}^E$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC problem if for any set of unsafe edges, $F'$, with $|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a feasible edge-set $F$ that minimizes $c(F) = \sum_{e \in F} c_e$.