Breaching 2-approximation for Forest Augmentation Problem

The basic goal of survivable network design is to build cheap networks that guarantee the connectivity of certain pairs of nodes despite the failure of a few edges or nodes. A celebrated result by Jain [Combinatorica'01] provides a 2-approximation for a wide class of these problems. However nothing better is known even for very basic special cases, raising the natural question whether any improved approximation factor is possible at all.
In this paper we address one of the most basic problems in this family for which 2 is still the best-known approximation factor, the Forest Augmentation Problem (FAP): given an undirected unweighted graph (that w.l.o.g. we can assume to be a forest) and a collection of extra edges (links), compute a minimum cardinality subset of links whose addition to the graph makes it 2-edge-connected. Several better-than-2 approximation algorithms are known for the special case where the input graph is a tree, a.k.a. the Tree Augmentation Problem (TAP), see e.g. [Grandoni, Kalaitzis, Zenklusen - STOC'18; Cecchetto, Traub, Zenklusen - STOC'21] and references therein. Recently this was achieved also for the weighted version of TAP [Traub, Zenklusen - FOCS'21], and for the k-connectivity generalization of TAP [Byrka, Grandoni, Jabal-Ameli - STOC'20; Cecchetto, Traub, Zenklusen - STOC'21]. These results heavily exploit the fact that the input graph is connected, a condition that does not hold in FAP.
In this paper we breach the 2-approximation barrier for FAP. Our result is based on two main ingredients. First, we describe a reduction to the Path Augmentation Problem (PAP), the special case of FAP where the input graph is a collection of disjoint paths. Our reduction is not approximation preserving, however it is sufficiently accurate to improve on a factor 2 approximation. Second, we present a better-than-2 approximation algorithm for PAP, an open problem on its own. Here we exploit a novel notion of implicit credits which might turn out to be helpful in future related work.