Breaching the 2-Approximation Barrier for Connectivity Augmentation: a Reduction to Steiner Tree

The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of the most basic problems in this area: given a $k$(-edge)-connected graph $G$ and a set of extra edges (links), select a minimum cardinality subset $A$ of links such that adding $A$ to $G$ increases its edge connectivity to $k+1$. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is $2$, and this can be achieved with multiple approaches (the first such result is in [Frederickson and J\'aj\'a'81]). It is known [Dinitz et al.'76] that CAP can be reduced to the case $k=1$, a.k.a. the Tree Augmentation Problem (TAP), for odd $k$, and to the case $k=2$, a.k.a. the Cactus Augmentation Problem (CacAP), for even $k$. Several better than $2$ approximation algorithms are known for TAP, culminating with a recent $1.458$ approximation [Grandoni et al.'18]. However, for CacAP the best known approximation is $2$. In this paper we breach the $2$ approximation barrier for CacAP, hence for CAP, by presenting a polynomial-time $2\ln(4)-\frac{967}{1120}+\epsilon<1.91$ approximation. Previous approaches exploit properties of TAP that do not seem to generalize to CacAP. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al.'14]. This reduction is not approximation preserving, and using the current best approximation factor for Steiner tree [Byrka et al.'13] as a black-box would not be good enough to improve on $2$. To achieve the latter goal, we ``open the box'' and exploit the specific properties of the instances of Steiner tree arising from CacAP.