Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants

k -center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Wi...

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Veröffentlicht in:Algorithmica Jg. 86; H. 8; S. 2557 - 2574
Hauptverfasser: Bandyapadhyay, Sayan, Friggstad, Zachary, Mousavi, Ramin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.08.2024
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Approximation
Clustering
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Design factors
Euclidean geometry
Euclidean space
Lower bounds
Mathematics of Computing
Polynomials
Theory of Computation
Time dependence
Euclidean
Parameterized algorithms
Hardness
Non-uniform
center
ISSN:0178-4617, 1432-0541
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Zusammenfassung:k -center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an O ( n log k ) + ( 1 / ϵ ) O ( 2 d k 1 - 1 / d log k ) -time ( 1 + ϵ ) -approximation for Euclidean k -center, where d is the dimension. We show for a closely related problem, k -supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k -center and k -supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k -center, called Non-uniform k -center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2 O ( k log k ) n 2 time 3-approximation for NUkC in general metrics, and a 2 O ( ( k log k ) / ϵ ) d n time ( 1 + ϵ ) -approximation for Euclidean NUkC. The latter time bound matches the bound for k -center.
Bibliographie:ObjectType-Article-1
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-024-01236-1