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

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.

💡 Research Summary

The paper tackles a highly relevant variant of the classic k‑center clustering problem that simultaneously incorporates demographic fairness constraints and robustness to outliers. In the “fair k‑center with outliers” setting, the data points are partitioned into t demographic groups, each with an upper bound k_i on how many centers may be chosen from that group, and up to z points may be discarded as outliers. Prior work only achieved a bi‑criteria FPT approximation: a 3‑approximation on the clustering radius while allowing (1+ε)·z outliers, but no true (unicriteria) approximation was known.

The authors introduce a fundamentally different approach that avoids the two‑phase “unconstrained‑solution‑then‑projection” paradigm, which breaks down when outliers are present because the unconstrained centers may correspond to points that are outliers in the optimal fair solution. Instead, they construct a fair solution from the start. Their method consists of two main components:

1. Reduction to a “colorful” instance.
   They first transform the original problem into a version where each demographic group is split into unit‑requirement groups (each allowing at most one center). Then, using a randomized color‑coding technique, they assign each unit group a color from a palette of exactly k colors, merging all groups of the same color into a single color class. A feasible solution to this colorful instance must pick exactly one center from each color class, which implicitly satisfies the original fairness constraints. The random coloring preserves the optimal cost with probability 2^{‑Ω(k)}; by enumerating all 2^{O(k)} color assignments (possible in FPT time) they obtain a deterministic guarantee that at least one reduced instance has the same optimum as the original.

2. Iterative ball‑finding algorithm for colorful k‑center with outliers.
   For a fixed colorful instance, the algorithm proceeds in at most k phases. In each phase it aims to “capture” one optimal cluster. It guesses the color of the center of the largest uncovered optimal cluster (at most k possibilities) and then repeatedly extracts the densest ball of radius OPT among points of that color. A structural trichotomy guarantees that one of three situations occurs: (a) a nearby ball contains a point from the target optimal cluster, so expanding it to radius 3·OPT fully covers that cluster; (b) the current ball already contains enough points to simulate the optimal cluster; or (c) two light balls together provide enough “free” points to simulate the cluster. In all cases the algorithm can certify that the chosen ball (or union of balls) covers at least as many points as the optimal cluster, respecting the outlier budget. A charging scheme maps each point of the optimal solution to a distinct point covered by the algorithm’s balls, ensuring that after k phases all non‑outlier points are covered and at most z points remain uncovered.

The algorithm runs in time 2^{O(k)}·poly(n), i.e., fixed‑parameter tractable with respect to the number of clusters k. The authors also show how the same framework extends to the fair k‑supplier problem (where centers must be chosen from a prescribed set) and to the more general fair‑range model, where each group has both lower and upper bounds. In each case they obtain a deterministic 3‑approximation with the same FPT running time.

Finally, the paper establishes a matching hardness result: achieving any approximation factor strictly better than 3 is W