BetterThan$\frac{4}{3}$Approximations for LeaftoLeaf Tree and Connectivity Augmentation
Abstract
The Connectivity Augmentation Problem (CAP) together with a wellknown 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 matchingbased method for the wellknown special case of leaftoleaf instances. Combining our work with prior techniques, we readily obtain a $(\frac{4}{3}+\epsilon)$approximation for LeaftoLeaf CAP by returning the better of our solution and one of an existing method. Prior to our work, a $\frac{4}{3}$guarantee was only known for LeaftoLeaf TAP instances on trees of height $2$. Moreover, when combining our technique with a recently introduced stack analysis approach, which is part of the abovementioned $1.393$approximation, we can further improve the approximation factor to $1.29$, obtaining for the first time a factor below $\frac{4}{3}$ for a nontrivial class of TAP/CAP instances.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.06944
 arXiv:
 arXiv:2204.06944
 Bibcode:
 2022arXiv220406944C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Optimization and Control