  We present approximation algorithms for several network design problems in
the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and
M\"uhlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021),
and IPCO 2020: pp. 13-26).
  Let $k\geq 1$, $p\geq 1$ 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 $S$ and a set of unsafe edges $U$, and nonnegative costs $c: E\to\Re$ on
the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC
problem if for any subset $F'$ of unsafe edges with $|F'|\leq q$, the subgraph
$(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a
feasible solution $F$ that minimizes $c(F) = \sum_{e \in F} c_e$. 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 $O(q\log|V|)$-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 $\min(k,2 u_{max})$-approximation
algorithm for the Cap-k-ECSS problem, where $u_{max}$ denotes the maximum
capacity of an edge.
