FPT approximation for capacitated clustering with outliers

Abstract

Clustering problems such as k-Median, and k-Means, are motivated from applications such as location planning, unsupervised learning among others. In many such applications, it is important to find the clustering of points that is not “skewed” in terms of the number of points, i.e., no cluster should contain too many points. This is often modeled by introducing capacity constraints on the sizes of clusters. In an orthogonal direction, another important consideration in the domain of clustering is how to handle the presence of outliers in the data. Indeed, the aforementioned clustering problems have been generalized in the literature to separately handle capacity constraints and outliers. However, to the best of our knowledge, there has been very little work on studying the approximability of clustering problems that can simultaneously handle capacity constraints as well as outliers.

We bridge this gap and initiate the study of the Capacitated k-Median with Outliers (CkMO) problem. In this problem, we want to cluster all except m outlier points into at most k clusters, such that (i) the clusters respect the capacity constraints, and (ii) the cost of clustering, defined as the sum of distances of each non-outlier point to its assigned cluster-center, is minimized.

We design the first constant-factor approximation algorithms for CkMO. In particular, our algorithm returns a $(3+ϵ)$-approximation for CkMO in general metric spaces that runs in time $f(k,m,ϵ)\cdot |I_m{|}^{O\left(1\right)}$, where $\left|I_m\right|$ denotes the input size.