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The Connectivity Augmentation Problem (CAP) together with a well-known special case thereof known as the Tree Augmentation Problem (TAP) are among the most basic Network Design problems. There has been a surge of interest recently to find approximation algorithms with guarantees below 2 for both TAP and CAP, culminating in the currently best approximation factor for both problems of 1.393 through quite sophisticated techniques.We present a new and arguably simple matching-based method for the well-known special case of leaf-to-leaf instances. Combining our work with prior techniques, we readily obtain a ( 4 /3 + ε)-approximation for Leaf-to-Leaf CAP by returning the better of our solution and one of an existing method. Prior to our work, a 4 /3-guarantee was only known for Leaf-to-Leaf TAP instances on trees of height 2. Moreover, when combining our technique with a recently introduced stack analysis approach, which is part of the above-mentioned 1.393-approximation, we can further improve the approximation factor to 1.29, obtaining for the first time a factor below 4 3 for a nontrivial class of TAP/CAP instances.

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 1.393approximation 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 1.91approximation 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.