Better-Than-$\frac{4}{3}$-Approximations for Leaf-to-Leaf Tree and Connectivity Augmentation

TL;DR

This paper introduces a simple matching-based approach for leaf-to-leaf instances in the Connectivity Augmentation Problem, achieving approximation guarantees below 1.33 and improving previous bounds for TAP and CAP.

Contribution

It presents a new matching-based method for leaf-to-leaf instances, leading to better approximation ratios for TAP and CAP, including the first sub-1.33 guarantee for nontrivial instances.

Findings

Achieved a (4/3 + ε)-approximation for Leaf-to-Leaf CAP.

Improved the approximation factor for certain TAP/CAP instances to 1.29.

Extended guarantees beyond previously known cases for leaf-to-leaf instances.

Abstract

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 $(\frac{4}{3}+\epsilon)$-approximation for Leaf-to-Leaf 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 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 $\frac{4}{3}$ for a nontrivial class of TAP/CAP instances.