Bisect and Conquer: Hierarchical Clustering via Max-Uncut Bisection

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Vaggos Chatziafratis, Grigory Yaroslavtsev, Euiwoong Lee, Konstantin Makarychev, Sara Ahmadian, Alessandro Epasto, Mohammad Mahdian ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:3121-3132, 2020.

Abstract

Hierarchical Clustering is an unsupervised data analysis method which has been widely used for decades. Despite its popularity, it had an underdeveloped analytical foundation and to address this, Dasgupta recently introduced an optimization viewpoint of hierarchical clustering with pairwise similarity information that spurred a line of work shedding light on old algorithms (e.g., Average-Linkage), but also designing new algorithms. Here, for the maximization dual of Dasgupta’s objective (introduced by Moseley-Wang), we present polynomial-time 42.46% approximation algorithms that use Max-Uncut Bisection as a subroutine. The previous best worst-case approximation factor in polynomial time was 33.6%, improving only slightly over Average-Linkage which achieves 33.3%. Finally, we complement our positive results by providing APX-hardness (even for 0-1 similarities), under the Small Set Expansion hypothesis.

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