We present improved approximation algorithms for some problems in the related areas of Flexible Graph Connectivity and Capacitated Network Design. 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 $F$. In the related Cap-$k$-ECSS problem, we are given a graph $G(V,E)$ whose edges have arbitrary integer capacities, and the goal is to find a minimum cost subset of edges $F$ such that the graph $G'(V,F)$ is $k$-edge connected. We obtain a $7$-approximation algorithm for the $(1,q)$-FGC problem that improves upon the previous best $(q+1)$-approximation. We also give an $O(\log{k})$-approximation algorithm for the Cap-$k$-ECSS problem, improving upon the previous best $O(\log{n})$-approximation whenever $k = o(n)$. Both these results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then during the rounding process utilizing an $O(1)$-approximation algorithm for the problem of covering small cuts. We also show that the the problem of covering small cuts inherently arises in another variant of $(p,q)$-FGC. Specifically, we show $O(1)$-approximate reductions between the $(2,q)$-FGC problem and the 2-Cover$\;$Small$\;$Cuts problem where each small cut needs to be covered twice.

翻译：本文针对柔性图连通性与容量网络设计相关领域中的若干问题，提出了改进的近似算法。在$(p,q)$-柔性图连通性问题（记作$(p,q)$-FGC）中，输入为一个图$G(V, E)$，其边集$E$被划分为安全边与不安全边；目标是找到最小成本的边集$F$，使得子图$G'(V, F)$在从$F$中移除任意$q$条不安全边后仍保持$p$-边连通。在相关的Cap-$k$-ECSS问题中，给定一个边具有任意整数容量的图$G(V,E)$，目标是找到最小成本的边子集$F$，使得图$G'(V,F)$是$k$-边连通的。我们为$(1,q)$-FGC问题提出了一个$7$-近似算法，改进了先前最佳的$(q+1)$-近似结果。同时，我们为Cap-$k$-ECSS问题给出了一个$O(\log{k})$-近似算法，当$k = o(n)$时，该结果优于先前最佳的$O(\log{n})$-近似。这两个结果均通过使用经背包覆盖不等式增强的自然线性规划松弛，并在舍入过程中利用覆盖小割问题的$O(1)$-近似算法获得。我们还证明，覆盖小割问题本质上也出现在$(p,q)$-FGC的另一种变体中。具体而言，我们展示了$(2,q)$-FGC问题与2-Cover$\;$Small$\;$Cuts问题之间的$O(1)$-近似归约，其中每个小割需要被覆盖两次。