# Approximation Algorithms for Cost-Balanced Clustering

Published in Preprint, 2019

Recommended citation: Amit Deshpande, Anand Louis, Deval Patel, Apoorv Singh. Approximation Algorithm for Cost-Balanced Clustering. In Submission. https://www.dropbox.com/s/r5uwemki3zfvvyb/min_max_km.pdf?dl=0

Clustering points in the Euclidean space is a fundamental problem in the theory of algorithms and in unsupervised learning. Various clustering objectives to quantify the quality of clustering have been proposed and studied; the $k$-means and $k$-median clustering objective are the most popular ones. In some cases, the $k$-means or the $k$-median objective may result in a few clusters of very large cost and many clusters of extremely small cost. Even when the optimal clusters are balanced in size, some of them may have a huge variance. This is undesirable for quantization or when we have a budget constraint on the cost of each cluster. Motivated by this, we study the cost-balanced $k$-means and the cost-balanced $k$-median problem. For a $k$-clustering $O_1, \ldots, O_k$ of a given set of $n$ points $X \subset \mathbb R^d$, we define its cost-balanced $k$-means cost as:

$\boxed{\mathcal K ({O_1, \ldots, O_k}) \stackrel{def}{=} \max_{l \in [k]} \sum_{x \in O_l} \left\lVert{x - \mu_l}\right\rVert^2, \qquad \textrm{where } \mu_l = \frac{1}{\left\lvert{O_l}\right\rvert} \sum_{x \in O_l} x ~}$

In other words, we want to minimize the cost of the heaviest cluster or balance the cost of each cluster. For any $\varepsilon > 0$, we give a randomized algorithm with running time $\mathcal O\big({2^{poly \big({k/\varepsilon}\big)} n d }\big)$ that gives a $(1+\varepsilon)$-approximation to the optimal cost-balanced $k$-means and the similarly defined optimal cost-balanced $k$-median clustering, using $k$ clusters, with a constant probability. We define a more general version of the $k$-median clustering and the cost-balanced $k$-median clustering, and we name them $\ell_p$ cost $k$-clustering and $\ell_p$ cost-balanced $k$-clustering, respectively. Given a black-box algorithm which gives a constant factor approximation to the $\ell_p$ cost $k$-clustering, we show a procedure that runs in time $poly(n,k,p)$ which gives a bi-criteria $\mathcal O\big({1/\varepsilon^{1/p}}\big)$-approximation to the optimal $\ell_p$ cost-balanced $k$-clustering, using $(1+\varepsilon)k$ clusters.