Breaching the 2Approximation Barrier for the Forest Augmentation Problem
Abstract
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 2approximation 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 bestknown approximation factor, the Forest Augmentation Problem (FAP): given an undirected unweighted graph (that w.l.o.g. is a forest) and a collection of extra edges (links), compute a minimum cardinality subset of links whose addition to the graph makes it 2edgeconnected. Several betterthan2 approximation algorithms are known for the special case where the input graph is a tree, a.k.a. the Tree Augmentation Problem (TAP). Recently this was achieved also for the weighted version of TAP, and for the kedgeconnectivity generalization of TAP. 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 2approximation 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 betterthan2 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.11799
 arXiv:
 arXiv:2112.11799
 Bibcode:
 2021arXiv211211799G
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics;
 Mathematics  Optimization and Control