Dependent randomized rounding for clustering and partition systems with knapsack constraints
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Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:22732283, 2020.
Abstract
Clustering problems are fundamental to unsupervised learning. There is an increased emphasis on \emph{fairness} in machine learning and AI; one representative notion of fairness is that no single demographic group should be overrepresented among the clustercenters. This, and much more general clustering problems, can be formulated with “knapsack" and “partition" constraints. We develop new randomized algorithms targeting such problems, and study two in particular: multiknapsack median and multiknapsack center. Our rounding algorithms give new approximation and pseudoapproximation algorithms for these problems. One key technical tool we develop and use, which may be of independent interest, is a new tail bound analogous to Feige (2006) for sums of random variables with unbounded variances. Such bounds are very useful in inferring properties of large networks using few samples.
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