Tight FPT Approximation for Socially Fair Clustering

In this work, we study the socially fair $k$-median/$k$-means problem. We are given a set of points $P$ in a metric space $\mathcal{X}$ with a distance function $d(.,.)$. There are $\ell$ groups: $P_1,\dotsc,P_{\ell} \subseteq P$. We are also given a set $F$ of feasible centers in $\mathcal{X}$. The goal in the socially fair $k$-median problem is to find a set $C \subseteq F$ of $k$ centers that minimizes the maximum average cost over all the groups. That is, find $C$ that minimizes the objective function $\Phi(C,P) \equiv \max_{j} \Big\{ \sum_{x \in P_j} d(C,x)/|P_j| \Big\}$, where $d(C,x)$ is the distance of $x$ to the closest center in $C$. The socially fair $k$-means problem is defined similarly by using squared distances, i.e., $d^{2}(.,.)$ instead of $d(.,.)$. The current best approximation guarantee for both the problems is $O\left( \frac{\log \ell}{\log \log \ell} \right)$ due to Makarychev and Vakilian [COLT 2021]. In this work, we study the fixed parameter tractability of the problems with respect to parameter $k$. We design $(3+\varepsilon)$ and $(9 + \varepsilon)$ approximation algorithms for the socially fair $k$-median and $k$-means problems, respectively, in FPT (fixed parameter tractable) time $f(k,\varepsilon) \cdot n^{O(1)}$, where $f(k,\varepsilon) = (k/\varepsilon)^{{O}(k)}$ and $n = |P \cup F|$. Furthermore, we show that if Gap-ETH holds, then better approximation guarantees are not possible in FPT time.