Parameterized Approximation for Robust Clustering in Discrete Geometric Spaces

TL;DR

This work advances parameterized approximations for Robust $(k,z)$-Clustering in geometric spaces by (i) surpassing general-metric barriers with a $3^z(1-\eta_0)$ FPT approximate algorithm in discrete high-dimensional Euclidean spaces, (ii) establishing a W[1]-hardness result for discrete $k$-Center in dimensions $\Theta(\log n)$, and (iii) delivering an EPAS for metrics with sub-logarithmic doubling dimension via coresets and ball-decomposition techniques. The core technical contributions include a strengthened assignment/projection lemma leveraging mid-point closure, a coreset framework for doubling metrics, and a careful hardness construction from Multi-Colored Independent Set. The results collectively map the FPT-approximation landscape for Robust $(k,z)$-Clustering in geometric spaces, showing both improved algorithmic prospects and sharp hardness boundaries, with practical impact for robust and fair clustering in high-dimensional data settings. Key mathematical features include the factor $3^z(1-\eta_0)$ in FPT time, the coreset size $(\tfrac{2^z}{\varepsilon})^{O(d)} kz \log n$, and the EPAS runtime $((\tfrac{2^z}{\varepsilon})^d k \log k)^{O(k)}$ in doubling metrics.

Abstract

We consider the well-studied Robust $(k, z)$-Clustering problem, which generalizes the classic $k$-Median, $k$-Means, and $k$-Center problems. Given a constant $z\ge 1$, the input to Robust $(k, z)$-Clustering is a set $P$ of $n$ weighted points in a metric space $(M,δ)$ and a positive integer $k$. Further, each point belongs to one (or more) of the $m$ many different groups $S_1,S_2,\ldots,S_m$. Our goal is to find a set $X$ of $k$ centers such that $\max_{i \in [m]} \sum_{p \in S_i} w(p) δ(p,X)^z$ is minimized. This problem arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness. For polynomial time computation, an approximation factor of $O(\log m/\log\log m)$ is known [Makarychev, Vakilian, COLT $2021$], which is tight under a plausible complexity assumption even in the line metrics. For FPT time, there is a $(3^z+ε)$-approximation algorithm, which is tight under GAP-ETH [Goyal, Jaiswal, Inf. Proc. Letters, 2023]. Motivated by the tight lower bounds for general discrete metrics, we focus on \emph{geometric} spaces such as the (discrete) high-dimensional Euclidean setting and metrics of low doubling dimension, which play an important role in data analysis applications. First, for a universal constant $η_0 >0.0006$, we devise a $3^z(1-η_{0})$-factor FPT approximation algorithm for discrete high-dimensional Euclidean spaces thereby bypassing the lower bound for general metrics. We complement this result by showing that even the special case of $k$-Center in dimension $Θ(\log n)$ is $(\sqrt{3/2}- o(1))$-hard to approximate for FPT algorithms. Finally, we complete the FPT approximation landscape by designing an FPT $(1+ε)$-approximation scheme (EPAS) for the metric of sub-logarithmic doubling dimension.

Figures (4)

• Figure 1: This example shows that the projection lemma is tight even for the $1$-dimensional Euclidean space. Let $o=0$ be the optimum facility located at the origin and serving client $p=1/2$. Let $b'=1$ be the facility in $B$ that serves $p$ and let $b=\sigma(o)=-1$ be the facility in $B$ nearest to $o$. We have $\textsf{OPT}=1/2$, which also equals the cost of $B$. However $\delta(p,\sigma(o))=3/2=2\times \delta(p,o) + 1\times \delta(p,b')$. Combining multiple such examples in orthogonal directions and sharing facility $b$ shows that the approximation ratio of the algorithm of Goyal and Jaiswal goyal2021tight approaches $3$ in the discrete Euclidean space.
• Figure 2: The midpoint of $b$ and $b"$ is shown by red dot, $||(b+b")/2 - o || \leq \frac{\alpha}{2}$ and thus $||\sigma(o)-o||\leq\alpha$.
• Figure 3: The dashed black circle depicts ${\sf ball}(o,1)$, while the dashed gray circles represent ${\sf ball}(o, 1-\omega)$ and ${\sf ball}(o,1+\omega)$. Regions $R_1$, $R_2$, $R_3$, and $R_4$ are outlined with green, yellow, purple, and blue borders respectively.
• Figure 5: An illustration of $D$ and the projection points of $p$, $q$ and $b'$.

Theorems & Definitions (30)

• Theorem 1.1: High-Dimensional Euclidean Space
• Theorem 1.2: Hardness in Discrete Euclidean Space
• Theorem 1.3: EPAS for Doubling Metric of Sub-Logarithmic Dimension
• Lemma 2.1: Projection Lemma
• Theorem 3.1: High-Dimensional Euclidean Space
• Lemma 3.1: Projection Lemma
• proof
• Lemma 3.1: Assignment Lemma
• Definition 3.1: Displacement Ratio
• Claim 3.1