Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part II

TL;DR

This paper improves the approximation ratio for the unweighted Tree Augmentation Problem by analyzing the Lasserre SDP relaxation and providing a combinatorial algorithm with guarantees close to the relaxation's integrality ratio.

Contribution

It establishes a bound on the integrality ratio of the Lasserre SDP relaxation for TAP and presents a combinatorial algorithm achieving an approximation within this bound.

Findings

Integrality ratio of SDP relaxation is at most 1.5 + ε.

A polynomial-time combinatorial algorithm achieves a (1.5 + ε)-approximation.

Extension of analysis from integral to fractional solutions using Lasserre system properties.

Abstract

In Part II, we study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum~of~Squares) system. We prove that the integrality ratio of an SDP relaxation (the Lasserre tightening of an LP relaxation) is $\leq \frac{3}{2}+\epsilon$, where $\epsilon>0$ can be any small constant. We obtain this result by designing a polynomial-time algorithm for TAP that achieves an approximation guarantee of ($\frac32+\epsilon$) relative to the SDP relaxation. The algorithm is combinatorial and does not solve the SDP relaxation, but our analysis relies on the SDP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Karlin, Mathieu and Nguyen (IPCO 2011).