Document Zbl 1522.68654

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V.</a>; <a href="/authors/pyatkin.artem-valerevich" title="Author Profile">Pyatkin, A. V.</a>; <a href="/authors/khandeev.vladimir-ilich" title="Author Profile">Khandeev, V. I.</a></div> <h2 class="title"> <strong>Quadratic Euclidean 1-mean and 1-median 2-clustering problem with constraints on the size of the clusters: complexity and approximability.</strong> <i>(English. Russian original)</i> <a class="label nowrap" href="/1522.68654">Zbl 1522.68654</a> </h2> <div class="source"> <a href="/serials/4210" title="Journal Profile">Proc. Steklov Inst. Math.</a> <a href="/?q=in%3A467299" title="Articles in this Issue">313, Suppl. 1, S117-S124 (2021)</a>; translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 25, No. 4, 69-78 (2019). </div> <div class="abstract">Summary: We consider the problem of partitioning a set of $N$ points in $d$-dimensional Euclidean space into two clusters minimizing the sum of the squared distances between each element and the center of the cluster to which it belongs. The center of the first cluster is its centroid (the geometric center). The center of the second cluster should be chosen among the points of the input set. We analyze the variant of the problem with given sizes (cardinalities) of the clusters; the sum of the sizes equals the cardinality of the input set. We prove that the problem is strongly NP-hard and there is no fully polynomial-time approximation scheme for it.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68U05" title="MSC2020">68U05</a> </td> <td class="space"> Computer graphics; computational geometry (digital and algorithmic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q17" title="MSC2020">68Q17</a> </td> <td class="space"> Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AEuclidean+space">Euclidean space</a>; <a href="/?q=ut%3Aclustering">clustering</a>; <a href="/?q=ut%3A2-partition">2-partition</a>; <a href="/?q=ut%3Aquadratic+variation">quadratic variation</a>; <a href="/?q=ut%3Acenter">center</a>; <a href="/?q=ut%3Acentroid">centroid</a>; <a href="/?q=ut%3Amedian">median</a>; <a href="/?q=ut%3Astrong+NP-hardness">strong NP-hardness</a>; <a href="/?q=ut%3Anonexistence+of+FPTAS">nonexistence of FPTAS</a>; <a href="/?q=ut%3Aapproximation-preserving+reduction">approximation-preserving reduction</a></div> <div class="software"> <h3>Software:</h3><a href="/software/42125">PRMLT</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 1522.68654" data-ciurl="/ci/07377486" data-biburl="/bibtex/07377486.bib" data-amsurl="/amsrefs/07377486.bib" data-xmlurl="/xml/07377486.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07377486.pdf" title="Zbl 1522.68654 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/10.1134/S0081543821030123" aria-label="DOI for “Quadratic Euclidean 1-mean and 1-median 2-clustering problem with constraints on the size of the clusters: complexity and approximability”" title="10.1134/S0081543821030123">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">Aloise, D.; Deshpande, A.; Hansen, P.; Popat, P., NP-hardness of Euclidean sum-of-squares clustering, Mach. 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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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