{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6682","name":"Dispersion in Unit Disks","abstract":"We present two new approximation algorithms with (improved) constant ratios for selecting $n$ points in $n$ unit disks such that the minimum pairwise distance among the points is maximized. \r\n\r\n(I) A very simple $O(n \\log{n})$-time algorithm with ratio $0.5110$ for disjoint unit disks. In combination with an algorithm of Cabello~\\cite{Ca07}, it yields a $O(n^2)$-time algorithm\r\nwith ratio of $0.4487$ for dispersion in $n$ not necessarily disjoint\r\nunit disks. \r\n\r\n(II) A more sophisticated LP-based algorithm with ratio $0.6495$ for\r\ndisjoint unit disks that uses a linear number of variables and\r\nconstraints, and runs in polynomial time. \r\nThe algorithm introduces a novel technique which combines linear\r\nprogramming and projections for approximating distances. \r\n\r\nThe previous best approximation ratio for disjoint unit disks was $\\frac{1}{2}$. Our results give a partial answer to an open question raised by Cabello~\\cite{Ca07}, who asked whether $\\frac{1}{2}$ could be improved.","keywords":["Dispersion problem","linear programming","approximation algorithm"],"author":[{"@type":"Person","name":"Dumitrescu, Adrian","givenName":"Adrian","familyName":"Dumitrescu"},{"@type":"Person","name":"Jiang, Minghui","givenName":"Minghui","familyName":"Jiang"}],"position":27,"pageStart":299,"pageEnd":310,"dateCreated":"2010-03-09","datePublished":"2010-03-09","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dumitrescu, Adrian","givenName":"Adrian","familyName":"Dumitrescu"},{"@type":"Person","name":"Jiang, Minghui","givenName":"Minghui","familyName":"Jiang"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2010.2464","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6208","volumeNumber":5,"name":"27th International Symposium on Theoretical Aspects of Computer Science","dateCreated":"2010-03-09","datePublished":"2010-03-09","editor":[{"@type":"Person","name":"Marion, Jean-Yves","givenName":"Jean-Yves","familyName":"Marion"},{"@type":"Person","name":"Schwentick, Thomas","givenName":"Thomas","familyName":"Schwentick"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6682","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6208"}}}