We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$.

中文翻译：

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缩小树和增强连接之间的差距：统一和更强大的方法

我们考虑连通性增强问题（CAP），这是可生存网络设计领域中的经典问题。它是关于以尽可能最便宜的方式将图形的边缘连接性提高一个单位。更精确地说，给定一个$ k $边连接的图$ G =（V，E）$和一组额外的边，任务是找到额外边的最小基数子集，这些额外边的边加上$ G $使得该图$（k + 1）$-边连接。如果$ k $是奇数，则已知该问题可以简化为Tree Augmentation问题（TAP）-即$ G $是一棵生成树-最近已经取得了重大进展，导致近似因子低于$ 1.5 $ （当前的最佳系数是$ 1.458 $）。但是，到目前为止，TAP的进展并没有延续到CAP。确实，直到最近Byrka，Grandoni Ameli（STOC 2020）通过提出一种与TAP的最新进展脱节的方法，给出了一种1.91美元的近似算法，从而获得了CAP低于2美元的第一个近似因子。我们首先通过介绍允许利用从TAP到CAP的见解和方法的技术来弥合TAP和CAP之间的鸿沟。然后，我们基于一种新的分析技术，采用一种新的方法来使逼近因子低于1.5美元。通过这些成分，我们获得了CAP约为1.393 $的算法，因此也获得了TAP。通过显着改善上述CAP的$ 1.91 $近似值以及Grandoni，Kalaitzis和Zenklusen对于TAP的$ 1.458 $的先前最佳近似值，这以统一的方式得出了这两个问题的当前最佳近似结果。 STOC 2018）。另外，