{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6878","name":"Balanced Interval Coloring","abstract":"We consider the discrepancy problem of coloring n intervals with k colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time O(n log n + kn log k) for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where $n$~tasks with given start- and endtimes have to be distributed among $k$~servers. Our results imply that this can be done ideally balanced.\r\n\r\nWhen generalizing to $d$-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any d >= 2 and any k >= 2 it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem.\r\n\r\nIn an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.","keywords":["Load balancing","discrepancy theory","NP-hardness"],"author":[{"@type":"Person","name":"Antoniadis, Antonios","givenName":"Antonios","familyName":"Antoniadis"},{"@type":"Person","name":"Hueffner, Falk","givenName":"Falk","familyName":"Hueffner"},{"@type":"Person","name":"Lenzner, Pascal","givenName":"Pascal","familyName":"Lenzner"},{"@type":"Person","name":"Moldenhauer, Carsten","givenName":"Carsten","familyName":"Moldenhauer"},{"@type":"Person","name":"Souza, Alexander","givenName":"Alexander","familyName":"Souza"}],"position":45,"pageStart":531,"pageEnd":542,"dateCreated":"2011-03-11","datePublished":"2011-03-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Antoniadis, Antonios","givenName":"Antonios","familyName":"Antoniadis"},{"@type":"Person","name":"Hueffner, Falk","givenName":"Falk","familyName":"Hueffner"},{"@type":"Person","name":"Lenzner, Pascal","givenName":"Pascal","familyName":"Lenzner"},{"@type":"Person","name":"Moldenhauer, Carsten","givenName":"Carsten","familyName":"Moldenhauer"},{"@type":"Person","name":"Souza, Alexander","givenName":"Alexander","familyName":"Souza"}],"copyrightYear":"2011","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2011.531","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6212","volumeNumber":9,"name":"28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)","dateCreated":"2011-03-10","datePublished":"2011-03-10","editor":[{"@type":"Person","name":"Schwentick, Thomas","givenName":"Thomas","familyName":"Schwentick"},{"@type":"Person","name":"D\u00fcrr, Christoph","givenName":"Christoph","familyName":"D\u00fcrr"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6878","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":"#volume6212"}}}