Tight FPT Approximations for Fair $k$-center with Outliers

TL;DR

This work delivers the first deterministic unicriterion FPT 3-approximation for fair $k$-center with outliers, running in time $2^{O(k\log k)}\cdot \mathrm{poly}(n)$, and proves that any improvement below 3 is W[2]-hard, establishing optimality under standard complexity assumptions. The approach directly constructs fair solutions via a novel iterative ball-finding framework and a reduction to colorful $k$-supplier with outliers through a color-coding-based, approximation-preserving transformation, avoiding projection-based paradigms. It further extends to fair $k$-supplier with outliers and to fair-range $k$-center with outliers by reducing to colorfully constrained instances, broadening the applicability of tight FPT approximations in constrained clustering. Together, these results close the gap between fairness, robustness, and efficient parameterized approximations for center-based clustering objectives.

Abstract

The $k$-center problem is a fundamental clustering objective that has been extensively studied in approximation algorithms. Recent work has sought to incorporate modern constraints such as fairness and robustness, motivated by biased and noisy data. In this paper, we study fair $k$-center with outliers, where centers must respect group-based representation constraints while up to $z$ points may be discarded. While a bi-criteria FPT approximation was previously known, no true approximation algorithm was available for this problem. We present the first deterministic $3$-approximation algorithm running in fixed-parameter tractable time parameterized by $k$. Our approach departs from projection-based methods and instead directly constructs a fair solution using a novel iterative ball-finding framework, based on a structural trichotomy that enables fixed-parameter approximation for the problem. We further extend our algorithm to fair $k$-supplier with outliers and to the more general fair-range setting with both lower and upper bounds. Finally, we show that improving the approximation factor below $3$ is $\mathrm{W[2]}$-hard, establishing the optimality of our results.

Figures (2)

• Figure 1: The rectangle represents the full point set, while the colored discs correspond to the optimal clusters, with the remaining area representing outliers. The relative sizes of the discs illustrate the densities of the clusters. The iterative ball-finding subroutine is invoked for the densest uncovered optimal cluster, which is green ($\pi^*_1$), and some balls constructed by the subroutine are depicted by dashed boundaries. Here, $B_1$ is a nearby ball for $\pi^*_1$, $B_2$ is a good dense ball for $\pi^*_1$, and $B_3$ and $B_4$ are light balls intersecting another uncovered cluster $\pi^*_3$, whose centers both lie within distance $2\mathrm{OPT}$ of the center of $\pi^*_3$.
• Figure 2: Flow of the iterative ball-finding algorithm for colorful $k$-center with outliers. Lightly shaded boxes correspond to the three cases of the structural trichotomy, each enabling the algorithm to place a radius-$3\mathrm{OPT}$ ball that settles one optimal cluster while preserving color constraints.

Theorems & Definitions (22)

• Theorem 1
• Corollary 2
• Theorem 3
• Definition 1: Fair $k$-Center with Outliers
• Definition 2: Fair $k$-Supplier with Outliers
• Definition 3: Colorful $k$-Supplier with Outliers
• Remark 1
• Remark 2
• Theorem 4
• Lemma 3