Document Zbl 1525.90418

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Res. Lett.</a> <a href="/?q=in%3A490932" title="Articles in this Issue">50, No. 6, 693-698 (2022)</a>. </div> <div class="abstract">Summary: In this paper, we investigate the weighted tree augmentation problem (TAP), where the goal is to augment a tree with a minimum cost set of edges such that the graph becomes two edge connected. First we show that in weighted TAP, we can restrict our attention to trees which are binary and where all the non-tree edges go between two leaves of the tree. We then give a top-down coloring algorithm that differs from known techniques for approximating TAP. The algorithm we describe always gives a 2-approximation starting from any feasible fractional solution to the natural tree cut covering LP. When the structure of the fractional solution is such that all the edges with non-zero weight are at least $\alpha$, then this algorithm achieves a $\frac{2}{1+\alpha}$-approximation. We also investigate a variant of TAP where every tree edge must belong to a cycle of length three (triangle) in the solution. We give a $\Theta (\log n)$-approximation algorithm for this problem in the weighted case in $n$-node graphs and a 4-approximation in the unweighted case.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7631937">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C35" title="MSC2020">90C35</a> </td> <td class="space"> Programming involving graphs or networks </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68W25" title="MSC2020">68W25</a> </td> <td class="space"> Approximation algorithms </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C05" title="MSC2020">05C05</a> </td> <td class="space"> Trees </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C22" title="MSC2020">05C22</a> </td> <td class="space"> Signed and weighted graphs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B10" title="MSC2020">90B10</a> </td> <td class="space"> Deterministic network models in operations research </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aapproximation+algorithm">approximation algorithm</a>; <a href="/?q=ut%3Anetwork+design">network design</a>; <a href="/?q=ut%3Aintegrality+gap">integrality gap</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1525.90418" data-ciurl="/ci/07631937" data-biburl="/bibtex/07631937.bib" data-amsurl="/amsrefs/07631937.bib" data-xmlurl="/xml/07631937.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07631937.pdf" title="Zbl 1525.90418 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.orl.2022.10.007" aria-label="DOI for “Coloring down: 3/2-approximation for special cases of the weighted tree augmentation problem”" title="10.1016/j.orl.2022.10.007">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1707.05240"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adjiashvili, David, Beating approximation factor two for weighted tree augmentation with bounded costs, ACM Trans. Algorithms, 15, 2, 1-26 (2018) · <a href="/1454.68085" class="nowrap">Zbl 1454.68085</a></td> </tr><tr> <td>[2]</td> <td class="space">Arora, Sanjeev; Sudan, Madhu, Improved low-degree testing and its applications, Combinatorica, 23, 3, 365-426 (2003) · <a href="/1101.68572" class="nowrap">Zbl 1101.68572</a></td> </tr><tr> <td>[3]</td> <td class="space">Carr, Robert; Vempala, Santosh, Randomized metarounding, (Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing (2000)), 58-62 · <a href="/1296.68163" class="nowrap">Zbl 1296.68163</a></td> </tr><tr> <td>[4]</td> <td class="space">Cecchetto, Federica; Traub, Vera; Zenklusen, Rico, Bridging the gap between tree and connectivity augmentation: unified and stronger approaches, (Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021)), 370-383 · <a href="/07765178" class="nowrap">Zbl 07765178</a></td> </tr><tr> <td>[5]</td> <td class="space">Cheriyan, Joseph; Gao, Zhihan, Approximating (unweighted) tree augmentation via lift-and-project, part i: stemless tap, Algorithmica, 80, 2, 530-559 (2018) · <a href="/1395.90232" class="nowrap">Zbl 1395.90232</a></td> </tr><tr> <td>[6]</td> <td class="space">Cheriyan, Joseph; Gao, Zhihan, Approximating (unweighted) tree augmentation via lift-and-project, part ii, Algorithmica, 80, 2, 608-651 (2018) · <a href="/1395.90233" class="nowrap">Zbl 1395.90233</a></td> </tr><tr> <td>[7]</td> <td class="space">Cheriyan, Joseph; Jordán, Tibor; Ravi, R., On 2-coverings and 2-packings of laminar families, (Algorithms-ESA’99 (1999)), 72 · <a href="/0945.05014" class="nowrap">Zbl 0945.05014</a></td> </tr><tr> <td>[8]</td> <td class="space">Cheriyan, Joseph; Karloff, Howard; Khandekar, Rohit; Könemann, Jochen, On the integrality ratio for tree augmentation, Oper. Res. Lett., 36, 4, 399-401 (2008) · <a href="/1155.90466" class="nowrap">Zbl 1155.90466</a></td> </tr><tr> <td>[9]</td> <td class="space">Cohen, Nachshon; Nutov, Zeev, A (1+ ln2)-approximation algorithm for minimum-cost 2-edge-connectivity augmentation of trees with constant radius, Theor. Comput. Sci., 489, 67-74 (2013) · <a href="/1294.05146" class="nowrap">Zbl 1294.05146</a></td> </tr><tr> <td>[10]</td> <td class="space">Even, Guy; Feldman, Jon; Kortsarz, Guy; Nutov, Zeev, A 1.8-approximation for augmenting edge-connectivity of a graph from 1 to 2, ACM Trans. Algorithms, 5, 2 (2009) · <a href="/1398.68670" class="nowrap">Zbl 1398.68670</a></td> </tr><tr> <td>[11]</td> <td class="space">Fiorini, Samuel; Groß, Martin; Könemann, Jochen; Sanità, Laura, Approximating weighted tree augmentation via Chvátal-Gomory cuts, (Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (2018), SIAM), 817-831 · <a href="/1403.68347" class="nowrap">Zbl 1403.68347</a></td> </tr><tr> <td>[12]</td> <td class="space">Frederickson, Greg N.; Jájá, Joseph, Approximation algorithms for several graph augmentation problems, SIAM J. Comput., 10, 2, 270-283 (1981) · <a href="/0461.05040" class="nowrap">Zbl 0461.05040</a></td> </tr><tr> <td>[13]</td> <td class="space">Fredrickson, Greg N.; Jájá, Joseph, On the relationship between the biconnectivity augmentation and traveling salesman problem, Theor. Comput. Sci., 19, 2, 189-201 (1982) · <a href="/0486.90082" class="nowrap">Zbl 0486.90082</a></td> </tr><tr> <td>[14]</td> <td class="space">Goldberg, Andrew V., Finding a maximum density subgraph (1984), EECS Department, University of California: EECS Department, University of California Berkeley, Technical Report UCB/CSD-84-171</td> </tr><tr> <td>[15]</td> <td class="space">Haddadan, Arash; Newman, Alantha, Efficient constructions of convex combinations for 2-edge-connected subgraphs on fundamental classes, Discrete Optim., 42, Article 100659 pp. (2021) · <a href="/1506.90258" class="nowrap">Zbl 1506.90258</a></td> </tr><tr> <td>[16]</td> <td class="space">Haddadan, Arash; Newman, Alantha; Ravi, R., Shorter tours and longer detours: uniform covers and a bit beyond, Math. Program., 185, 1, 245-273 (2021) · <a href="/1458.90546" class="nowrap">Zbl 1458.90546</a></td> </tr><tr> <td>[17]</td> <td class="space">Jain, Kamal, A factor 2 approximation algorithm for the generalized Steiner network problem, Combinatorica, 21, 1, 39-60 (2001) · <a href="/1107.68533" class="nowrap">Zbl 1107.68533</a></td> </tr><tr> <td>[18]</td> <td class="space">Khuller, Samir; Thurimella, Ramakrishna, Approximation algorithms for graph augmentation, J. Algorithms, 14, 2, 214-225 (1993) · <a href="/0764.68120" class="nowrap">Zbl 0764.68120</a></td> </tr><tr> <td>[19]</td> <td class="space">Kortsarz, Guy; Nutov, Zeev, A simplified 3/2 ratio approximation algorithm for the tree augmentation problem, ACM Trans. Algorithms, 12, 2, 23 (2016) · <a href="/1398.68679" class="nowrap">Zbl 1398.68679</a></td> </tr><tr> <td>[20]</td> <td class="space">Kortsarz, Guy; Nutov, Zeev, Lp-relaxations for tree augmentation, Discrete Appl. Math., 239, 94-105 (2018) · <a href="/1382.05016" class="nowrap">Zbl 1382.05016</a></td> </tr><tr> <td>[21]</td> <td class="space">Nutov, Zeev, On the tree augmentation problem, (25th Annual European Symposium on Algorithms (ESA 2017), Leibniz International Proceedings in Informatics (LIPIcs) (2017)), Article 61 pp. · <a href="/1442.68181" class="nowrap">Zbl 1442.68181</a></td> </tr><tr> <td>[22]</td> <td class="space">Ravi, R., Steiner Trees and Beyond: Approximation Algorithms for Network Design (1994), Brown University, PhD thesis</td> </tr><tr> <td>[23]</td> <td class="space">Traub, Vera; Zenklusen, Rico, Local search for weighted tree augmentation and Steiner tree, (Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2022), SIAM), 3253-3272 · <a href="/07883709" class="nowrap">Zbl 07883709</a></td> </tr><tr> <td>[24]</td> <td class="space">Traub, Vera; Zenklusen, Riko, A better-than-2 approximation for weighted tree augmentation, (Proceedings of the 62nd IEEE FOCS (2021)) (2021), CoRR</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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