Approximating (Unweighted) Tree Augmentation via Lift-and-Project (Part 0: $1.8+\epsilon$ approximation for (Unweighted) TAP)

TL;DR

This paper presents a new approximation algorithm for the unweighted Tree Augmentation Problem using the Lasserre (Sum of Squares) system, achieving a guarantee of (1.8+ε) relative to an SDP relaxation, matching previous combinatorial bounds.

Contribution

It introduces a novel analysis of fractional solutions of the Lasserre system for TAP, extending combinatorial guarantees to fractional solutions.

Findings

Achieves (1.8+ε) approximation guarantee for unweighted TAP.

Extends combinatorial analysis to fractional solutions via Lasserre system properties.

Matches previous combinatorial approximation bounds using SDP relaxation.

Abstract

We study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. We prove an approximation guarantee of ($1.8+\epsilon$) relative to an SDP relaxation, which matches the combinatorial approximation guarantee of Even, Feldman, Kortsarz and Nutov in ACM TALG (2009), where $\epsilon>0$ is a constant. We generalize the combinatorial analysis of integral solutions of Even, et al., to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Rothvo{\ss} (arXiv:1111.5473, 2011) and Karlin, Mathieu and Nguyen (IPCO 2011).