{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8719","name":"Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers","abstract":"We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {L_i}_{i in F} and a set D of clients located in a common metric space {c(i,j)}_{i,j in F union D}, and bounds k, m. A feasible solution is a pair (S subseteq F, sigma: D -> S union {out}), where sigma specifies the client assignments, such that |S| <=k, |sigma^{-1}(i)| >= L_i for all i in S, and |sigma^{-1}(out)| <= m. In the lower-bounded min-sum-of-radii with outliers P (LBkSRO) problem, the objective is to minimize sum_{i in S} max_{j in sigma^{-1})i)}, and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize max_{i in S} max_{j in sigma^{-1})i)} c(i,j).\r\n\r\nWe obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| <= k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well.\r\n\r\nWe obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.","keywords":["Approximation algorithms","facililty-location problems","primal-dual method","Lagrangian relaxation","k-center problems","minimizing sum of radii"],"author":[{"@type":"Person","name":"Ahmadian, Sara","givenName":"Sara","familyName":"Ahmadian"},{"@type":"Person","name":"Swamy, Chaitanya","givenName":"Chaitanya","familyName":"Swamy"}],"position":69,"pageStart":"69:1","pageEnd":"69:15","dateCreated":"2016-08-23","datePublished":"2016-08-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ahmadian, Sara","givenName":"Sara","familyName":"Ahmadian"},{"@type":"Person","name":"Swamy, Chaitanya","givenName":"Chaitanya","familyName":"Swamy"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.69","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/dx.doi.org\/10.4230\/LIPIcs.STACS.2014.251","isPartOf":{"@type":"PublicationVolume","@id":"#volume6258","volumeNumber":55,"name":"43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)","dateCreated":"2016-08-23","datePublished":"2016-08-23","editor":[{"@type":"Person","name":"Chatzigiannakis, Ioannis","givenName":"Ioannis","familyName":"Chatzigiannakis"},{"@type":"Person","name":"Mitzenmacher, Michael","givenName":"Michael","familyName":"Mitzenmacher"},{"@type":"Person","name":"Rabani, Yuval","givenName":"Yuval","familyName":"Rabani"},{"@type":"Person","name":"Sangiorgi, Davide","givenName":"Davide","familyName":"Sangiorgi"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8719","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":"#volume6258"}}}