We present improved approximation algorithms for some problems in the related areas of Capacitated Network Design and Flexible Graph Connectivity. In the Cap-$k$-ECSS problem, we are given a graph $G=(V,E)$ whose edges have non-negative costs and positive integer capacities, and the goal is to find a minimum-cost edge-set $F$ such that every non-trivial cut of the graph $G'=(V,F)$ has capacity at least $k$. We present an $O(\log k)$-approximation algorithm for the Cap-$k$-ECSS problem, asymptotically improving upon the previous best approximation ratio of $\min(O(\log n),\; O(k))$ whenever $\log(k)=o(\log n)$, where $n$ denotes $|V|$. (See section 1, for a detailed discussion.) In the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, the input is a graph $G(V, E)$ where $E$ is partitioned into safe and unsafe edges, and the goal is to find a minimum cost set of edges $F$ such that the subgraph $G'(V, F)$ remains $p$-edge connected upon removal of any $q$ unsafe edges from $F$. We design a $7$-approximation algorithm for the $(1,q)$-FGC problem, improving on the previous best approximation ratio of $(q+1)$. Both of our results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then, during the rounding process, utilizing a recent $O(1)$-approximation algorithm for the Cover$\;$Small$\;$Cuts problem. In the latter problem, the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than a specified value is covered by a link. We also show that the problem of covering small cuts inherently arises in another variant of $(p,q)$-FGC. Specifically, we give Cook reductions that preserve approximation ratios within $O(1)$ factors between the $(2,q)$-FGC problem and the 2-Cover$\;$Small$\;$Cuts problem; in the latter problem, each small cut needs to be covered by two links.

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