On Small-Depth Tree Augmentations

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 $2 - \frac{1}{2^{k-1}}$\mathrm{.}For2-and3-leveltrees,theseratiosare$\frac32$and$\frac74$ respectively. Our proofs are constructive and yield polynomial-time approximation algorithms with matching guarantees.