Dependent randomized rounding for clustering and partition systems with knapsack constraints

TL;DR

This paper introduces new randomized algorithms for clustering problems with knapsack constraints, addressing fairness and other considerations, and provides novel tail bounds for sums of unbounded variance variables.

Contribution

It develops a dependent randomized rounding technique for clustering with knapsack and partition constraints, including multi-knapsack median and center problems, with improved approximation guarantees.

Findings

New approximation algorithms for multi-knapsack median and center.

A novel tail bound for sums of random variables with unbounded variances.

Enhanced understanding of clustering under fairness and resource constraints.

Abstract

Clustering problems are fundamental to unsupervised learning. There is an increased emphasis on fairness in machine learning and AI; one representative notion of fairness is that no single demographic group should be over-represented among the cluster-centers. 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: multi-knapsack median and multi-knapsack center. Our rounding algorithms give new approximation and pseudo-approximation algorithms for these problems. One key technical tool, 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 can be useful in inferring properties of large networks using few samples.