NDLI: On Optimal Weighted Balanced Clusterings: Gravity Bodies and Power Diagrams

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Brieden, Andreas Gritzmann, Peter
Copyright Year	2012
Abstract	We study weighted clustering problems in Minkowski spaces under balancing constraints with a view towards separation properties. First, we introduce the gravity polytopes and more general gravity bodies that encode all feasible clusterings and indicate how they can be utilized to develop efficient approximation algorithms for quite general, hard to compute objective functions. Then we show that their extreme points correspond to strongly feasible power diagrams, certain specific cell complexes, whose defining polyhedra contain the clusters, respectively. Further, we characterize strongly feasible centroidal power diagrams in terms of the local optima of some ellipsoidal function over the gravity polytope. The global optima can also be characterized in terms of the separation properties of the corresponding clusterings.
Starting Page	415
Ending Page	434
Page Count	20
File Format	PDF
ISSN	08954801
DOI	10.1137/110832707
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	26
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-04-04
Access Restriction	Subscribed
Subject Keyword	gravity polytope Computer graphics; computational geometry clustering Computational learning theory power diagram Clustering Combinatorial optimization gravity body weighted clustering constrained clustering
Content Type	Text
Resource Type	Article
Subject	Mathematics

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