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Data Structure
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Centerbased Clustering under Perturbation Stability
Centerbased Clustering under Perturbation Stability,Computing Research Repository,Pranjal Awasthi,Avrim Blum,Or Sheffet
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Centerbased Clustering under Perturbation Stability
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Citations: 1
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Pranjal Awasthi
,
Avrim Blum
,
Or Sheffet
Optimal clustering is a notoriously hard task. Recently, Bilu and Linial suggested an approach towards understanding the complexity of clustering instances which arise in practice. They argue that such instances are often stable to perturbations in the
metric space
and give an
efficient algorithm
for clustering instances which are stable to perturbations of size $O(n^{1/2})$ for maxcut based clustering. In addition, they conjecture that instances stable to as little as O(1) perturbations should be solvable in polynomial time. In this paper we prove that this conjecture is true for any centerbased clustering objective (such as $k$median, $k$means, and $k$center), i.e., we can efficiently find the optimal clustering assuming only stability to {\em constant}magnitude perturbations of the underlying metric. Specifically, we show that for centerbased clustering instances which are stable to 3 perturbations, a combination of the popular singlelinkage heuristic together with dynamic programming will find the optimal clustering. In fact, our algorithm works for a weaker notion of stability we call {\em $\alpha$center stability}, and we show that for this weaker notion, beating the factor of 3 is NPhard. We also study how the notion of stability to perturbations relates to other definitions of clustering stability that have been proposed recently.
Journal:
Computing Research Repository  CORR
, vol. abs/1009.3, 2010
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References
(10)
Are stable instances easy?
(
Citations: 7
)
Yonatan Bilu
,
Nathan Linial
Journal:
Computing Research Repository  CORR
, vol. abs/0906.3, pp. 332341, 2009
A ConstantFactor Approximation Algorithm for the kMST Problem
(
Citations: 51
)
Avrim Blum
,
R. Ravi
,
Santosh Vempala
Journal:
Journal of Computer and System Sciences  JCSS
, vol. 58, no. 1, pp. 101108, 1999
Approximation schemes for clustering problems
(
Citations: 63
)
Wenceslas Fernandez de la Vega
,
Marek Karpinski
,
Claire Kenyon
,
Yuval Rabani
Conference:
ACM Symposium on Theory of Computing  STOC
, pp. 5058, 2003
Computers and Intractability: A Guide to the Theory of NPCompleteness
(
Citations: 19196
)
Michael Randolph Garey
,
David S. Johnson
Conference:
Artificial Evolution  AE
, 1979
Greedy Strikes Back: Improved Facility Location Algorithms
(
Citations: 148
)
Sudipto Guha
,
Samir Khuller
Journal:
Journal of Algorithms  JAL
, vol. 31, no. 1, pp. 228248, 1999
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Citations
(1)
Approximate Nash Equilibria under Stability Conditions
(
Citations: 1
)
MariaFlorina Balcan
,
Mark Braverman
Journal:
Computing Research Repository  CORR
, vol. abs/1008.1, 2010