## Abstract

In Part I, we study a special case of the unweighted *Tree Augmentation Problem* (TAP) via the Lasserre (Sum of Squares) system. In the special case, we forbid so-called stems; these are a particular type of subtree configuration. For stemless TAP, we prove that the integrality ratio of an SDP relaxation (the Lasserre tightening of an LP relaxation) is \(\le \frac{3}{2}+\epsilon \), where \(\epsilon >0\) can be any small constant. We obtain this result by designing a polynomial-time algorithm for stemless TAP that achieves an approximation guarantee of \(\left( \frac{3}{2}+\epsilon \right) \) 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 et al. (Integer programming and combinatoral optimization (IPCO), Lecture Notes in Computer Science, vol 6655. Springer, Berlin/Heidelberg, pp 301–314, 2011). Also, we present an example of stemless TAP such that the approximation guarantee of \(\frac{3}{2}\) is tight for the algorithm. In Part II of this paper, we extend the methods of Part I to prove the same results relative to the same SDP relaxation for TAP.

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## Notes

- 1.
In other words, a link covers every tree-edge in the unique path of

*T*between the ends of the link. - 2.
A preliminary version of our paper with a weaker approximation guarantee for TAP relative to the same SDP relaxation was circulated widely in July 2014, see [1], and it led to subsequent publications by others, e.g., [15], but we prefer to avoid discussion on these matters and we leave it to the subsequent publications.

- 3.
Although we defined \(\textsc {Las}_t({ LP}_0)\) to be a set of vectors in \(\mathbb {R}^{2^{[|E|]}}\), in what follows, we abuse the notation and take \(\textsc {Las}_t({ LP}_0)\) to be the projection on the subspace indexed by the singleton sets; thus, we take \(\textsc {Las}_t({ LP}_0)\) to be a set of vectors in \(\mathbb {R}^{{|E|}}\).

- 4.
Recall that \(\mathcal {R}\) is the set of original non-leaf nodes of

*T*; thus \(V(T'_v)\cap \mathcal {R}\) denotes the set of nodes of \(T'_v\) excluding all leaves and all compound nodes.

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## Acknowledgements

We thank André Linhares for many discussions. We thank several other colleagues who read preliminary drafts and gave us insightful comments.

Funding was provided by Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN-2014-04351).

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Cheriyan, J., Gao, Z. Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part I: Stemless TAP.
*Algorithmica* **80, **530–559 (2018). https://doi.org/10.1007/s00453-016-0270-4

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### Keywords

- Approximation algorithms
- Lift-and-Project
- Linear programming
- Semidefinite programming
- Network design
- Algorithmic aspects of networks
- Graph connectivity
- Connectivity augmentation