Document Zbl 07978138

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Comput. Sci.</a> <a href="/?q=in%3A523325" title="Articles in this Issue">1029, Article ID 115050, 7 p. (2025)</a>. </div> <div class="abstract">Summary: We reconsider the min-max clustered cycle cover (MM-CCC) problem, which is described as follows. Given an undirected complete graph $G = (V, E; w)$ with a positive integer $k$, where the vertex set $V$ is partitioned into $h$ clusters $V_1, \ldots, V_h$, and $w : E \to \mathbb{R}^+$ is an edge-weight function satisfying the triangle inequality, it is asked to find $k$ cycles such that they traverse all vertices and the vertices in each cluster are required to be traversed consecutively. The objective is to minimize the weight of the maximum weight cycle. We propose a strongly polynomial time 16-approximation algorithm for the MM-CCC problem. The result improves the previous algorithm in terms of running time.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68R10" title="MSC2020">68R10</a> </td> <td class="space"> Graph theory (including graph drawing) in computer science </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%3A90C27" title="MSC2020">90C27</a> </td> <td class="space"> Combinatorial optimization </td> </tr><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%3A90C59" title="MSC2020">90C59</a> </td> <td class="space"> Approximation methods and heuristics in mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acombinatorial+optimization">combinatorial optimization</a>; <a href="/?q=ut%3Acycle+cover">cycle cover</a>; <a href="/?q=ut%3Aclustered">clustered</a>; <a href="/?q=ut%3Aapproximation+algorithm">approximation algorithm</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 07978138" data-ciurl="/ci/07978138" data-biburl="/bibtex/07978138.bib" data-amsurl="/amsrefs/07978138.bib" data-xmlurl="/xml/07978138.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07978138.pdf" title="Zbl 07978138 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.tcs.2024.115050" aria-label="DOI for “A strongly polynomial time approximation algorithm for the min-max clustered cycle cover problem”" title="10.1016/j.tcs.2024.115050">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Christofides, N., Worst-case analysis of a new heuristic for the travelling salesman problem, 1976, Graduate School of Industrial Administration, Carnegie Mellon University, Report 388</td> </tr><tr> <td>[2]</td> <td class="space">Karlin, A. R.; Klein, N.; Oveis Gharan, S., A (slightly) improved approximation algorithm for metric TSP, (Proceedings of the Fifty-Third Annual ACM SIGACT Symposium on Theory of Computing (STOC), 2021, ACM: ACM New York, USA), 32-45 · <a href="/07765151" class="nowrap">Zbl 07765151</a></td> </tr><tr> <td>[3]</td> <td class="space">Golden, B. L.; Raghavan, S.; Wasil, E. A., The Vehicle Routing Problem: Latest Advances and New Challenges, 2008, Springer: Springer New York · <a href="/1142.90004" class="nowrap">Zbl 1142.90004</a></td> </tr><tr> <td>[4]</td> <td class="space">Laporte, G., Fifty years of vehicle routing, Transp. Sci., 43, 408-416, 2009</td> </tr><tr> <td>[5]</td> <td class="space">Toth, P.; Vigo, D., Vehicle Routing: Problems, Methods and Applications, 2014, MOS-SIAM: MOS-SIAM Philadelphia · <a href="/1305.90012" class="nowrap">Zbl 1305.90012</a></td> </tr><tr> <td>[6]</td> <td class="space">Campbell, A. M.; Vandenbussche, D.; Hermann, W., Routing for relief efforts, Transp. 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Sci., 654, 45-58, 2016 · <a href="/1359.90125" class="nowrap">Zbl 1359.90125</a></td> </tr><tr> <td>[11]</td> <td class="space">Laporte, G.; Palekar, U., Some applications of the clustered travelling salesman problem, J. Oper. Res. Soc., 53, 972-976, 2002 · <a href="/1139.90425" class="nowrap">Zbl 1139.90425</a></td> </tr><tr> <td>[12]</td> <td class="space">Arkin, E.; Hassin, R.; Klein, L., Restricted delivery problems on a network, Networks, 29, 205-216, 1997 · <a href="/0882.90030" class="nowrap">Zbl 0882.90030</a></td> </tr><tr> <td>[13]</td> <td class="space">Guttmann-Beck, N.; Hassin, R.; Khuller, S.; Raghavachari, B., Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem, Algorithmica, 28, 422-437, 2000 · <a href="/0963.68226" class="nowrap">Zbl 0963.68226</a></td> </tr><tr> <td>[14]</td> <td class="space">Bao, X.; Liu, Z., An improved approximation algorithm for the clustered traveling salesman problem, Inf. Process. Lett., 112, 908-910, 2012 · <a href="/1248.68551" class="nowrap">Zbl 1248.68551</a></td> </tr><tr> <td>[15]</td> <td class="space">Bao, X.; Xu, L.; Yu, W.; Song, W., Approximation algorithms for the min-max clustered k-traveling salesmen problems, Theor. Comput. Sci., 933, 60-66, 2022 · <a href="/1540.90229" class="nowrap">Zbl 1540.90229</a></td> </tr><tr> <td>[16]</td> <td class="space">Hoogeveen, J. A., Analysis of Christofides’ heuristic: some paths are more difficult than cycles, Oper. Res. Lett., 10, 291-295, 1991 · <a href="/0748.90071" class="nowrap">Zbl 0748.90071</a></td> </tr><tr> <td>[17]</td> <td class="space">Eiselt, H. A.; Gendreau, M.; Laporte, G., Arc routing problems, part II: the rural postman problem, Oper. Res., 43, 399-414, 1995 · <a href="/0853.90042" class="nowrap">Zbl 0853.90042</a></td> </tr><tr> <td>[18]</td> <td class="space">Yu, W.; Liu, Z., Better approximability results for min-max tree/cycle/path cover problems, J. Comb. Optim., 37, 563-578, 2019 · <a href="/1422.90048" class="nowrap">Zbl 1422.90048</a></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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