{"id":324395,"date":"2016-11-18T16:27:37","date_gmt":"2016-11-19T00:27:37","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=324395"},"modified":"2018-10-16T20:41:08","modified_gmt":"2018-10-17T03:41:08","slug":"non-uniform-k-center-problem","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/non-uniform-k-center-problem\/","title":{"rendered":"The Non-Uniform k-Center Problem"},"content":{"rendered":"

In this paper, we introduce and study the Non-Uniform k-Center problem (NUkC). Given a finite metric space (<\/span>X<\/span>,<\/span>d<\/span>)<\/span><\/span><\/span><\/span> and a collection of balls of radii {<\/span>r<\/span>1<\/span><\/span>\u2265<\/span>\u22ef<\/span>\u2265<\/span>r<\/span>k<\/span><\/span>}<\/span><\/span><\/span><\/span>, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation \u03b1<\/span><\/span><\/span><\/span>, such that the union of balls of radius \u03b1<\/span>\u22c5<\/span>r<\/span>i<\/span><\/span><\/span><\/span><\/span> around the i<\/span><\/span><\/span><\/span>th center covers all the points in X<\/span><\/span><\/span><\/span>. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds.
\nThe NUkC problem generalizes the classic k<\/span><\/span><\/span><\/span>-center problem when all the k<\/span><\/span><\/span><\/span> radii are the same (which can be assumed to be 1<\/span><\/span><\/span><\/span> after scaling). It also generalizes the k<\/span><\/span><\/span><\/span>-center with outliers (kCwO) problem when there are k<\/span><\/span><\/span><\/span> balls of radius 1<\/span><\/span><\/span><\/span> and \u2113<\/span><\/span><\/span><\/span> balls of radius 0<\/span><\/span><\/span><\/span>. There are 2<\/span><\/span><\/span><\/span>-approximation and 3<\/span><\/span><\/span><\/span>-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years.
\nWe first observe that no O<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span>-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an (<\/span>O<\/span>(<\/span>1<\/span>)<\/span>,<\/span>O<\/span>(<\/span>1<\/span>)<\/span>)<\/span><\/span><\/span><\/span>-bi-criteria approximation result: we give an O<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span>-approximation to the optimal dilation, however, we may open \u0398<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span> centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal 2<\/span><\/span><\/span><\/span>-approximation to the kCwO problem improving upon the long-standing 3<\/span><\/span><\/span><\/span>-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this paper, we introduce and study the Non-Uniform k-Center problem (NUkC). Given a finite metric space (X,d) and a collection of balls of radii {r1\u2265\u22ef\u2265rk}, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation \u03b1, such that the union of balls of radius […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, Rome, 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