Clustering What Matters in Constrained Settings

TL;DR

The paper addresses outlier-constrained clustering for $k$-median and $k$-means by introducing an approximation-preserving reduction to the corresponding outlier-free problems. The core method uses a D^z-sampling based reduction that creates $q=f(k,m,\varepsilon)=\left( \frac{k+m}{\varepsilon}\right)^{O(m)}$ outlier-free instances and combines solutions via a randomized selection with high probability guarantees. When an $\alpha$-approximation for the outlier-free problem is available, the approach yields an $\alpha(1+\varepsilon)$-approximation for the outlier version, with quantified running time that scales with $q$ and the underlying subroutines. The work extends to labelled constrained clustering and general metric spaces, providing the first FPT approximation guarantees for several constrained outlier problems and broadening the applicability of reduction-based techniques in constrained clustering. This advance enables near-optimal solutions in settings with hard constraints (e.g., capacitated, fair, coloured) and a limited number of outliers, with implications for robust clustering in diverse data regimes.

Abstract

Constrained clustering problems generalize classical clustering formulations, e.g., $k$-median, $k$-means, by imposing additional constraints on the feasibility of clustering. There has been significant recent progress in obtaining approximation algorithms for these problems, both in the metric and the Euclidean settings. However, the outlier version of these problems, where the solution is allowed to leave out $m$ points from the clustering, is not well understood. In this work, we give a general framework for reducing the outlier version of a constrained $k$-median or $k$-means problem to the corresponding outlier-free version with only $(1+\varepsilon)$-loss in the approximation ratio. The reduction is obtained by mapping the original instance of the problem to $f(k,m, \varepsilon)$ instances of the outlier-free version, where $f(k, m, \varepsilon) = \left( \frac{k+m}{\varepsilon}\right)^{O(m)}$. As specific applications, we get the following results: - First FPT (in the parameters $k$ and $m$) $(1+\varepsilon)$-approximation algorithm for the outlier version of capacitated $k$-median and $k$-means in Euclidean spaces with hard capacities. - First FPT (in the parameters $k$ and $m$) $(3+\varepsilon)$ and $(9+\varepsilon)$ approximation algorithms for the outlier version of capacitated $k$-median and $k$-means, respectively, in general metric spaces with hard capacities. - First FPT (in the parameters $k$ and $m$) $(2-δ)$-approximation algorithm for the outlier version of the $k$-median problem under the Ulam metric. Our work generalizes the known results to a larger class of constrained clustering problems. Further, our reduction works for arbitrary metric spaces and so can extend clustering algorithms for outlier-free versions in both Euclidean and arbitrary metric spaces.

Figures (3)

• Figure 1: An example 2-dimensional instance with ($k=3; m = 3; F = C$), where the red stars are the optimal outliers. The reduction algorithm finds a set $C$ of $k+m = 6$ centers ( shown as blue triangles). It then $D^z$-samples $O(m \log{m})$ points with respect to center set $C$, which guarantees that the faraway outliers ( see red stars shaded with green circles) are found. The outliers near $C$ ( see the top red star) are not discovered this way. So, we find a suitable "replacement or proxy" ( see the point shaded with green square) for such outliers by setting up a $b$-matching problem to locate suitable points that are close to the centers in $C$. The instance for the outlier-free version is obtained by removing a suitable subset of proxies and faraway outliers from the point set ( see figure on the right). The key technicality lies in showing that designating proxies as outliers does not increase the cost too much.
• Figure 2: The optimal outliers with closest center as $c_j$ ( see red stars) are denoted by $X_{N, j}^{opt}$. Since we cannot distinguish them from other points near $c_j$, we find their proxies $\hat{X}_j$ ( see points shaded green). Even though we show these sets as disjoint in the diagram, they may contain common points. We will designate $\hat{X}_j$ as the outlier points. This replacement of optimal outliers with their proxies may cause a loss. However, this loss can be bounded by the sum of distances between an optimal outlier and its image as per a one-to-one mapping $\mu$ ( see dotted arrows) between $X_{N, j}^{opt}$ and $\hat{X}_j$.
• Figure 3: We want to designate the proxies as outliers instead of their pre-images ( as per the mapping $\mu$ defined in Figure \ref{['fig:lemma-1']}). The penalty of this replacement will not be too much, as per Lemma \ref{['lem:mu']}. However, there is an issue with this plan if a proxy point in $\hat{X}_i$ is also an optimal outlier in $X^{opt}_{N, j}$ for $i \neq j$ ( see the star shaded with a green circle). In this case, we modify the one-to-one mapping $\mu$ to $\hat{\mu}$ by tracing the mapping $\mu$ starting from an optimal outlier to a non-outlier ( see star on the left to point on the right). We map the extreme points to each other and map the intermediate points to themselves ( see yellow dashed lines). The penalty of this mapping will now depend on the distance between the extreme points, but that can be bounded by applying the approximate triangle inequality along the path.

Theorems & Definitions (17)

• theorem 1: Main Theorem
• theorem 2: Main Theorem: labelled version
• Definition 3
• Claim 1
• proof
• Claim 2
• proof
• Claim 3
• proof
• lemma 1