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.

翻译：我们针对容量约束网络设计与柔性图连通性相关领域中的若干问题提出了改进的近似算法。在Cap-k-ECSS问题中，给定图$G=(V,E)$，其边具有非负成本和正整数容量，目标是找到最小成本边集$F$，使得图$G'=(V,F)$中每个非平凡割的容量至少为$k$。我们给出了Cap-k-ECSS问题的$O(\log k)$-近似算法，当$\log(k)=o(\log n)$时（其中$n$表示$|V|$），该结果渐近改进了此前最优近似比$\min(O(\log n),\; O(k))$（详见第1节讨论）。在$(p,q)$-柔性图连通性问题（记为$(p,q)$-FGC）中，输入为图$G(V, E)$，其边集$E$划分为安全边与不安全边，目标是找到最小成本边集$F$，使得子图$G'(V, F)$在移除$F$中任意$q$条不安全边后仍保持$p$-边连通。我们为$(1,q)$-FGC问题设计了$7$-近似算法，改进了此前最优近似比$(q+1)$。这两项成果均通过使用背包覆盖不等式强化的自然线性规划松弛实现，并在舍入过程中利用最近提出的Cover$\;$Small$\;$Cuts问题$O(1)$-近似算法。在Cover$\;$Small$\;$Cuts问题中，目标是找到最小成本链路集，使得每个容量小于指定值的非平凡割被某条链路覆盖。我们还证明，覆盖小割问题天然产生于$(p,q)$-FGC的另一变体中。具体而言，我们给出了$(2,q)$-FGC问题与2-Cover$\;$Small$\;$Cuts问题之间保持$O(1)$因子近似比的Cook归约，在后一问题中每个小割需要被两条链路覆盖。