FPT Approximation for Socially Fair Clustering

• URL: http://arxiv.org/abs/2106.06755v1
• Date: Sat, 12 Jun 2021 11:53:18 GMT
• Title: FPT Approximation for Socially Fair Clustering
• Authors: Dishant Goyal and Ragesh Jaiswal
• Abstract summary: We are given a set of points $P$ in a metric space $mathcalX$ with a distance function $d(.,.)$. 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. In this work, we design $(5+varepsilon)$ and $(33 + varepsilon)$ approximation algorithms for the socially fair $k$-median and $k$-means
• Score: 0.38073142980733
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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