A unified approach to approximation algorithms for bottleneck problems
Journal of the ACM (JACM)
Optimal algorithms for approximate clustering
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Cluster analysis and mathematical programming
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
Approximation algorithms for clustering to minimize the sum of diameters
Nordic Journal of Computing
Clustering to minimize the sum of cluster diameters
Journal of Computer and System Sciences - STOC 2001
Online clustering with variable sized clusters
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
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We consider the problems of set partitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost function is given by an oracle, and we assume that it satisfies some natural structural constraints. That is, we assume that the cost function is monotone, the cost of a singleton is zero, and we assume that for all S ∩ S′ &neq; ∅ the following holds c(S) + c(S′) ≥ c(S ∪ S′). For the problem of minimizing the maximum cost of a cluster we present a (2k − 1)-approximation algorithm for k ≥ 3, a 2-approximation algorithm for k = 2, and we also show a lower bound of k on the performance guarantee of any polynomial-time algorithm. For the problem of minimizing the total cost of all the clusters, we present a 2-approximation algorithm for the case where k is a fixed constant, a (4k − 3)-approximation where k is unbounded, and we show a lower bound of 2 on the approximation ratio of any polynomial-time algorithm. Our lower bounds do not depend on the common assumption that P &neq; NP.