On Small-Depth Tree Augmentations

Abstract

We study the Weighted Tree Augmentation Problem for general link costs. We show that the integrality gap of the ODD-LP relaxation for the (weighted) Tree Augmentation Problem for a $k$-level tree instance is at most $2-\frac{1}{2^{k-1}}$. For 2- and 3-level trees, these ratios are $\frac{3}{2}$ and $\frac{7}{4}$ respectively. Our proofs are constructive and yield polynomial-time approximation algorithms with matching guarantees.