Document Zbl 1433.68612

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New York, NY: Association for Computing Machinery (ACM). 126-137 (2019). </div> <div class="abstract"></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7160495">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7160495">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A90B35" title="MSC2020">90B35</a> </td> <td class="space"> Deterministic scheduling theory in operations research </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B80" title="MSC2020">90B80</a> </td> <td class="space"> Discrete location and assignment </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C27" title="MSC2020">90C27</a> </td> <td class="space"> Combinatorial optimization </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aapproximation+algorithm">approximation algorithm</a>; <a href="/?q=ut%3Aclustering">clustering</a>; <a href="/?q=ut%3Aload+balancing">load balancing</a>; <a href="/?q=ut%3Aminimum+norm+optimization">minimum norm optimization</a>; <a href="/?q=ut%3Aordered+optimization">ordered optimization</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1427.68366">Zbl 1427.68366</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 1433.68612" data-ciurl="/ci/07160495" data-biburl="/bibtex/07160495.bib" data-amsurl="/amsrefs/07160495.bib" data-xmlurl="/xml/07160495.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07160495.pdf" title="Zbl 1433.68612 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/10.1145/3313276.3316322" aria-label="DOI for “Approximation algorithms for minimum norm and ordered optimization problems”" title="10.1145/3313276.3316322">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1811.05022"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">S. Ahmadian, A. Norouzi-Fard, O. Svensson, and J. Ward. 2017. Better guarantees for k-means and Euclidean k-median by primal-dual algorithms. In Proc., FOCS. · <a href="/1450.90005" class="nowrap">Zbl 1450.90005</a></td> </tr><tr> <td>[2]</td> <td class="space">S. Alamdari and D. B. Shmoys. 2017. A bicriteria approximation algorithm for the k-center and k-median problems. In Proc., WAOA. · <a href="/1442.68261" class="nowrap">Zbl 1442.68261</a></td> </tr><tr> <td>[3]</td> <td class="space">N. Alon, Y. Azar, G. Woeginger, and T. Yadid. 1998. Approximation schemes for scheduling on parallel machines. Journal of Scheduling 1, 1 (1998), 55-66. · <a href="/0909.90168" class="nowrap">Zbl 0909.90168</a></td> </tr><tr> <td>[4]</td> <td class="space">A. Andoni, H. Nguyen, A. Nikolov, I. Razenshteyn, and E. Waingarten. 2017.</td> </tr><tr> <td>[5]</td> <td class="space">Approximate near neighbors for general symmetric norms. In Proc., STOC. 10.1145/3055399.3055418 · <a href="/1370.68066" class="nowrap">Zbl 1370.68066</a></td> </tr><tr> <td>[6]</td> <td class="space">A. Aouad and D. Segev. 2018. The ordered k-median problem: surrogate models and approximation algorithms. Math. Programming (2018), 1-29. · <a href="/1456.68229" class="nowrap">Zbl 1456.68229</a></td> </tr><tr> <td>[7]</td> <td class="space">B. Awerbuch, Y. Azar, E. Grove, M. Kao, P. Krishnan, and J. S. Vitter. 1995. Load balancing in the L p norm. In Proc., FOCS. · <a href="/0938.68509" class="nowrap">Zbl 0938.68509</a></td> </tr><tr> <td>[8]</td> <td class="space">Y. Azar and A. Epstein. 2005.</td> </tr><tr> <td>[9]</td> <td class="space">Convex programming for scheduling unrelated parallel machines. In Proc., STOC. 10.1145/1060590.1060639 · <a href="/1192.90059" class="nowrap">Zbl 1192.90059</a></td> </tr><tr> <td>[10]</td> <td class="space">Y. Azar, L. Epstein, Y. Richter, and G. J. Woeginger. 2004. All-norm approximation algorithms. J. Algorithms 52, 2 (2004), 120-133. 10.1016/j.jalgor.2004.02.003 · <a href="/1072.68130" class="nowrap">Zbl 1072.68130</a></td> </tr><tr> <td>[11]</td> <td class="space">N. Bansal, R. Khandekar, and V. Nagarajan. 2009. Additive guarantees for degreebounded directed network design. SICOMP 39, 4 (2009), 1413-1431. · <a href="/1206.68366" class="nowrap">Zbl 1206.68366</a></td> </tr><tr> <td>[12]</td> <td class="space">R. Bhatia. 2013.</td> </tr><tr> <td>[13]</td> <td class="space">Matrix analysis. Vol. 169. Springer.</td> </tr><tr> <td>[14]</td> <td class="space">J. Błasiok, V. Braverman, S. Chestnut, R. Krauthgamer, and L. Yang. 2017. Streaming symmetric norms via measure concentration. In Proc., STOC. 10.1145/3055399.3055424 · <a href="/1369.68205" class="nowrap">Zbl 1369.68205</a></td> </tr><tr> <td>[15]</td> <td class="space">J. Byrka, T. Pensyl, B. Rybicki, A. Srinivasan, and K. Trinh. 2014. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proc., SODA. · <a href="/1371.90073" class="nowrap">Zbl 1371.90073</a></td> </tr><tr> <td>[16]</td> <td class="space">J. Byrka, K. Sornat, and J. Spoerhase. 2018. Constant-factor approximation for ordered k-median. In Proc., STOC. 10.1145/3188745.3188930 · <a href="/1427.68366" class="nowrap">Zbl 1427.68366</a></td> </tr><tr> <td>[17]</td> <td class="space">R. Carr and S. Vempala. 2002. Randomized metarounding. Random Structures Algorithms 20, 3 (2002), 343-352. 10.1002/rsa.10033 · <a href="/1005.90050" class="nowrap">Zbl 1005.90050</a></td> </tr><tr> <td>[18]</td> <td class="space">D. Chakrabarty and C. Swamy. 2018.</td> </tr><tr> <td>[19]</td> <td class="space">Interpolating between k-median and kcenter: Approximation Algorithms for Ordered k-median. In Proc., ICALP. · <a href="/1499.68401" class="nowrap">Zbl 1499.68401</a></td> </tr><tr> <td>[20]</td> <td class="space">M. Charikar, S. Guha, É. Tardos, and D. B. Shmoys. 2002.</td> </tr><tr> <td>[21]</td> <td class="space">A constant-factor approximation algorithm for the k-median problem. J. Comput. System Sci. 65, 1 (2002), 129-149. 10.1006/jcss.2002.1882 · <a href="/1023.90037" class="nowrap">Zbl 1023.90037</a></td> </tr><tr> <td>[22]</td> <td class="space">M. Charikar and S. Li. 2012. A dependent LP-rounding approach for the k-median problem. In Proc., ICALP. 10.1007/978-3-642-31594-7_17 · <a href="/1272.90020" class="nowrap">Zbl 1272.90020</a></td> </tr><tr> <td>[23]</td> <td class="space">A. Goel and A. Meyerson. 2006.</td> </tr><tr> <td>[24]</td> <td class="space">Simultaneous optimization via approximate majorization for concave profits or convex costs. Algorithmica 44, 4 (2006), 301-323. · <a href="/1113.90070" class="nowrap">Zbl 1113.90070</a></td> </tr><tr> <td>[25]</td> <td class="space">D. Golovin, A. Gupta, A. Kumar, and K. Tangwongsan. 2008.</td> </tr><tr> <td>[26]</td> <td class="space">All-norms and all- ℓ p -norms approximation algorithms. In Proc., FSTTCS.</td> </tr><tr> <td>[27]</td> <td class="space">T. F. Gonzalez. 1985. Clustering to Minimize the Maximum Intercluster Distance. Theoretical Computer Science 38 (1985), 293 - 306. · <a href="/0567.62048" class="nowrap">Zbl 0567.62048</a></td> </tr><tr> <td>[28]</td> <td class="space">A. Gupta and K. Tangwongsan. 2008. Simpler analyses of local search algorithms for facility location. arXiv preprint arXiv:0809.2554 (2008).</td> </tr><tr> <td>[29]</td> <td class="space">G. H. Hardy, J. E. Littlewood, and G. Pólya. 1934. · <a href="/60.0169.01" class="nowrap">JFM 60.0169.01</a></td> </tr><tr> <td>[30]</td> <td class="space">Inequalities. Camb. Univ Press.</td> </tr><tr> <td>[31]</td> <td class="space">D. S. Hochbaum and D. B. Shmoys. 1985.</td> </tr><tr> <td>[32]</td> <td class="space">A Best Possible Heuristic for the k-Center Problem. Math. Oper. Res. 10, 2 (1985), 180-184. 10.1287/moor.10.2.180 · <a href="/0565.90015" class="nowrap">Zbl 0565.90015</a></td> </tr><tr> <td>[33]</td> <td class="space">K. Jain and V. V. Vazirani. 2001. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. Journal of the ACM (JACM) 48, 2 (2001), 274-296. 10.1145/375827.375845 · <a href="/1138.90417" class="nowrap">Zbl 1138.90417</a></td> </tr><tr> <td>[34]</td> <td class="space">K. Jansen and L. Rohwedder. 2017.</td> </tr><tr> <td>[35]</td> <td class="space">On the configuration-LP of the restricted assignment problem. In Proc., SODA. · <a href="/1415.90041" class="nowrap">Zbl 1415.90041</a></td> </tr><tr> <td>[36]</td> <td class="space">I. Kabiljo, B. Karrer, M. Pundir, S. Pupyrev, and A. Shalita. 2017. Social hash partitioner: a scalable distributed hypergraph partitioner. Proc., Very Large Databases (VLDB) Endowment 10, 11 (2017), 1418-1429. 10.14778/3137628.3137650</td> </tr><tr> <td>[37]</td> <td class="space">A. Kumar and J. Kleinberg. 2006. Fairness measures for resource allocation. SIAM Journal on Computing (SICOMP) 36, 3 (2006), 657-680. · <a href="/1120.90028" class="nowrap">Zbl 1120.90028</a></td> </tr><tr> <td>[38]</td> <td class="space">V. S. Kumar, Madhav V Marathe, Srinivasan Parthasarathy, and Aravind Srinivasan. 2009. A unified approach to scheduling on unrelated parallel machines. Journal of the ACM (JACM) 56, 5 (2009), 28. 10.1145/1552285.1552289 · <a href="/1325.90044" class="nowrap">Zbl 1325.90044</a></td> </tr><tr> <td>[39]</td> <td class="space">G. Laporte, S. Nickel, and F. S. da Gama. 2015.</td> </tr><tr> <td>[40]</td> <td class="space">Location Science. Springer. · <a href="/1512.90002" class="nowrap">Zbl 1512.90002</a></td> </tr><tr> <td>[41]</td> <td class="space">L. C. Lau, R. Ravi, and M. Singh. 2011.</td> </tr><tr> <td>[42]</td> <td class="space">Iterative methods in combinatorial optimization. Vol. 46. Cambridge University Press. · <a href="/1248.68565" class="nowrap">Zbl 1248.68565</a></td> </tr><tr> <td>[43]</td> <td class="space">J. Lenstra, D. Shmoys, and E. Tardos. 1990. Approximation algorithms for scheduling unrelated parallel machines. Math. Programming 46, 1-3 (1990), 259-271. 10.1007/BF01585745 · <a href="/0715.90063" class="nowrap">Zbl 0715.90063</a></td> </tr><tr> <td>[44]</td> <td class="space">S. Li and O. Svensson. 2016. Approximating k-median via pseudo-approximation. SIAM Journal on Computing (SICOMP) 45, 2 (2016), 530-547. · <a href="/1338.90346" class="nowrap">Zbl 1338.90346</a></td> </tr><tr> <td>[45]</td> <td class="space">K. Makarychev and M. Sviridenko. 2014. Solving optimization problems with diseconomies of scale via decoupling. In Proc., FOCS. 10.1109/FOCS.2014.67 · <a href="/1426.90261" class="nowrap">Zbl 1426.90261</a></td> </tr><tr> <td>[46]</td> <td class="space">D. B. Shmoys and E. Tardos. 1993. An approximation algorithm for the generalized assignment problem. Mathematical programming 62, 1-3 (1993), 461-474. · <a href="/0804.90077" class="nowrap">Zbl 0804.90077</a></td> </tr><tr> <td>[47]</td> <td class="space">O. Svensson. 2012. Santa Claus schedules jobs on unrelated machines. SIAM J. Comput. 41, 5 (2012), 1318-1341. · <a href="/1257.68083" class="nowrap">Zbl 1257.68083</a></td> </tr><tr> <td>[48]</td> <td class="space">A. Tamir. 2001. The k-centrum multi-facility location problem. Discrete Applied Mathematics 109, 3 (2001), 293-307. Abstract 1 Introduction 2 Technical overview, organization 3 Preliminaries 4 Sparsifying weights 5 Reducing min-norm optimization to Min-Max ordered optimization 6 Proxy costs 7 Our approach for min-max ordered optimization 8 k-Clustering 8.1 LP Relaxation and Deterministic Oblivious Rounding 9 Load balancing 9.1 Linear Programming Relaxation References · <a href="/0986.90018" class="nowrap">Zbl 0986.90018</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. 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