On Small-Depth Tree Augmentations

TL;DR

This paper investigates the Weighted Tree Augmentation Problem for small-depth trees, providing new bounds on integrality gaps and developing polynomial-time approximation algorithms with proven guarantees.

Contribution

It establishes tight bounds on the integrality gap for k-level trees and offers constructive algorithms with matching approximation guarantees.

Findings

Integrality gap for k-level trees is at most 2 - 1/2^{k-1}.

For 2-level trees, the ratio is 3/2; for 3-level trees, it is 7/4.

Constructive proofs lead to polynomial-time algorithms with these guarantees.

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 $\frac32$ and $\frac74$ respectively. Our proofs are constructive and yield polynomial-time approximation algorithms with matching guarantees.