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

We study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. We prove an approximation guarantee of (1.8 + ) relative to an SDP relaxation, which matches the combinatorial approximation guarantee of Even, Feldman, Kortsarz and Nutov in ACM TALG (2009), where > 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 Rothvoß (arXiv:1111.5473, 2011) and Karlin, Mathieu and Nguyen (IPCO 2011). This is a manuscript from July 2014 that was widely circulated but was not posted on the web. It is being posted for the sake of archiving/referencing. The results have been subsumed by arXiv:1507.01309, our manuscript from July 2015, that proves an approximation guarantee of (1.5 + ) for (unweighted) TAP via a relaxation. The presentation of the weaker result here may be of interest because it is significantly shorter and simpler than arXiv:1507.01309. To the best of our knowledge, this was the earliest manuscript to prove an upper-bound less than 2 for the integrality ratio for a relaxation of TAP. (Although approximation guarantees less than 2 for (unweighted) TAP had been published earlier, those guarantees were proved w.r.t. an integral optimal solution and not w.r.t. a relaxation.) The original manuscript also had a proof of a (1.75 + ) approximation guarantee but that part has been omitted because the details are difficult; that result has been subsumed by arXiv:1507.01309. All our results use the same relaxation, namely, a Lasserre tightening of a simple LP relaxation. ar X iv :1 60 4. 00 70 8v 1 [ cs .D S] 4 A pr 2 01 6 APPROXIMATING (UNWEIGHTED) TREE AUGMENTATION VIA LIFT-AND-PROJECT JOSEPH CHERIYAN AND ZHIHAN GAO Abstract. We study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. We prove an approximation guarantee of (1.8 + ) relative to an SDP relaxation, which matches the combinatorial approximation guarantee of Even, Feldman, Kortsarz and Nutov in ACM TALG (2009), where > 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 Rothvoß (arXiv:1111.5473, 2011) and Karlin, Mathieu and Nguyen (IPCO 2011). We study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. We prove an approximation guarantee of (1.8 + ) relative to an SDP relaxation, which matches the combinatorial approximation guarantee of Even, Feldman, Kortsarz and Nutov in ACM TALG (2009), where > 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 Rothvoß (arXiv:1111.5473, 2011) and Karlin, Mathieu and Nguyen (IPCO 2011).