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 $X$ with a distance function $d(.,.)$. There are ℓ groups: $P_1,\dots ,P_{\ell }\subseteq P$. We are also given a set F of feasible centers in $X$. 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 $\Phi (C,P)\equiv {\max }_j\left\{{\sum }_{x\in P_j}d\right(C,x)/|P_j\left|\right\}$, 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 of 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 $(3+\epsilon )$ and $(9+\epsilon )$ approximation algorithms for the socially fair k-median and k-means problems, respectively, in FPT (fixed-parameter tractable) time $f(k,\epsilon )\cdot n^{O\left(1\right)}$, where $f(k,\epsilon )={(k/\epsilon )}^{O\left(k\right)}$ and $n=|P\cup F|$. The algorithms are randomized and succeed with a probability of at least $(1-\frac{1}{n})$. Furthermore, we show that if $W\left[2\right]\ne \mathrm{FPT}$, then better approximation guarantees are not possible in FPT time.