Constant factor approximations for Lower and Upper bounded Clusterings

TL;DR

This paper introduces a framework for approximating clustering problems with specified lower and upper bounds on cluster sizes, achieving constant factor approximations with controlled violations, thus aiding in privacy and analysis constraints.

Contribution

It provides the first constant factor approximation algorithms for lower and upper bounded clustering problems, including k-median and facility location, with bounded violations on upper bounds.

Findings

Achieves constant factor approximation for LUkM and LUkFL with bounded violations.

Improves upper bound violation for LUFL compared to previous work.

Provides approximation algorithms for LUkC and LUkS with uniform bounds.

Abstract

Clustering is one of the most fundamental problem in Machine Learning. Researchers in the field often require a lower bound on the size of the clusters to maintain anonymity and upper bound for the ease of analysis. Specifying an optimal cluster size is a problem often faced by scientists. In this paper, we present a framework to obtain constant factor approximations for some prominent clustering objectives, with lower and upper bounds on cluster size. This enables scientists to give an approximate cluster size by specifying the lower and the upper bounds for it. Our results preserve the lower bounds but may violate the upper bound a little. %{GroverGD21_LBUBFL_Cocoon} to $2$. %namely, $k$ Center (LUkC) and $k$ Median (LUkM) problem. We study the problems when either of the bounds is uniform. We apply our framework to give the first constant factor approximations for LUkM and its generalization, $k$-facility location problem (LUkFL), with $\beta+1$ factor violation in upper bounds where $\beta$ is the violation of upper bounds in solutions of upper bounded $k$-median and $k$-facility location problems respectively. We also present a result on LUkC with uniform upper bounds and, its generalization, lower and (uniform) upper bounded $k$ supplier problem (LUkS). The approach also gives a result on lower and upper bounded facility location problem (LUFL), improving upon the upper bound violation of $5/2$ due to Gupta et al. We also reduce the violation in upper bounds for a special case when the gap between the lower and upper bounds is not too small.