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 of 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 $\Phi(C,P) \equiv \max_{j} \sum_{x \in P_j} d(C,x)/|P_j|$, 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 $d(.,.)$. In this work, we design $(5+\varepsilon)$ and $(33 + \varepsilon)$ approximation algorithms for the socially fair $k$-median and $k$-means problems, respectively. For the parameters: $k$ and $\ell$, the algorithms have an FPT (fixed parameter tractable) running time of $f(k,\ell,\varepsilon) \cdot n$ for $f(k,\ell,\varepsilon) = 2^{{O}(k \, \ell/\varepsilon)}$ and $n = |P \cup F|$. We also study a special case of the problem where the centers are allowed to be chosen from the point set $P$, i.e., $P \subseteq F$. For this special case, our algorithms give better approximation guarantees of $(4+\varepsilon)$ and $(18+\varepsilon)$ for the socially fair $k$-median and $k$-means problems, respectively. Furthermore, we convert these algorithms to constant pass log-space streaming algorithms. Lastly, we show FPT hardness of approximation results for the problem with a small gap between our upper and lower bounds.

中文翻译：

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社会公平聚类的 FPT 近似

在这项工作中，我们研究了社会公平 $k$-median/$k$-means 问题。我们在具有距离函数 $d(.,.)$ 的度量空间 $\mathcal{X}$ 中给定一组点 $P$。有 $\ell$ 组：$P_1,\dotsc,P_{\ell} \subseteq P$。我们还给出了 $\mathcal{X}$ 中可行中心的集合 $F$。社会公平 $k$-median 问题的目标是找到一组 $C\subseteq F$ 的 $k$ 中心，使所有组的最大平均成本最小。也就是说，找到最小化目标函数 $\Phi(C,P) \equiv \max_{j} \sum_{x \in P_j} d(C,x)/|P_j|$ 的 $C$，其中 $d (C,x)$ 是 $x$ 到 $C$ 中最近的中心的距离。社会公平 $k$-means 问题的定义类似，使用平方距离，即 $d^{2}(.,.)$ 而不是 $d(.,.)$。在这项工作中，我们分别为社会公平的 $k$-median 和 $k$-means 问题设计了 $(5+\varepsilon)$ 和 $(33+\varepsilon)$ 近似算法。对于参数：$k$ 和 $\ell$，算法的 FPT（固定参数易处理）运行时间为 $f(k,\ell,\varepsilon) \cdot n$ for $f(k,\ell, \varepsilon) = 2^{{O}(k \, \ell/\varepsilon)}$ 和 $n = |P \cup F|$。我们还研究了允许从点集 $P$ 中选择中心的问题的一个特殊情况，即 $P \subseteq F$。对于这种特殊情况，我们的算法分别为社会公平的 $k$-median 和 $k$-means 问题提供了更好的近似保证 $(4+\varepsilon)$ 和 $(18+\varepsilon)$。此外，我们将这些算法转换为常量传递日志空间流算法。最后，