Abstract
In the Tree Augmentation Problem (TAP) the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T+F is 2edgeconnected. The best approximation ratio known for TAP is 1.5. In the more general Weighted TAP problem, F should be of minimum weight. Weighted TAP admits several 2approximation algorithms w.r.t. the standard cutLP relaxation. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than 2 is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LPrelaxation with integrality gap better than 2, which is an old open problem even for TAP. In this paper we introduce two different LPrelaxations, and for each of them give a simple algorithm that computes a feasible solution for TAP of size at most 7/4 times the optimal LP value. This gives some hope to break the ratio 2 for the weighted case.
BibTeX  Entry
@InProceedings{kortsarz_et_al:LIPIcs:2016:6636,
author = {Guy Kortsarz and Zeev Nutov},
title = {{LPRelaxations for Tree Augmentation}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
pages = {13:113:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770187},
ISSN = {18688969},
year = {2016},
volume = {60},
editor = {Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6636},
URN = {urn:nbn:de:0030drops66366},
doi = {10.4230/LIPIcs.APPROXRANDOM.2016.13},
annote = {Keywords: Tree Augmentation; LPrelaxation; Laminar family; Approximation algorithms}
}
Keywords: 

Tree Augmentation; LPrelaxation; Laminar family; Approximation algorithms 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016) 
Issue Date: 

2016 
Date of publication: 

06.09.2016 