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- W4288049772 abstract "We study the problem of fair $k$-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation $k$-median problem, we are given a set of points $X$ in a metric space. Each point $xin X$ belongs to one of $ell$ groups. Further, we are given fair representation parameters $alpha_j$ and $beta_j$ for each group $jin [ell]$. We say that a $k$-clustering $C_1, cdots, C_k$ fairly represents all groups if the number of points from group $j$ in cluster $C_i$ is between $alpha_j |C_i|$ and $beta_j |C_i|$ for every $jin[ell]$ and $iin [k]$. The goal is to find a set $mathcal{C}$ of $k$ centers and an assignment $phi: Xrightarrow mathcal{C}$ such that the clustering defined by $(mathcal{C}, phi)$ fairly represents all groups and minimizes the $ell_1$-objective $sum_{xin X} d(x, phi(x))$. We present an $O(log k)$-approximation algorithm that runs in time $n^{O(ell)}$. Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both $k$ and $ell$. We also consider an important special case of the problem where $alpha_j = beta_j = frac{f_j}{f}$ and $f_j, f in mathbb{N}$ for all $jin [ell]$. For this special case, we present an $O(log k)$-approximation algorithm that runs in $(kf)^{O(ell)}log n + poly(n)$ time." @default.
- W4288049772 created "2022-07-27" @default.
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- W4288049772 date "2022-02-02" @default.
- W4288049772 modified "2023-10-16" @default.
- W4288049772 title "Fair Representation Clustering with Several Protected Classes" @default.
- W4288049772 doi "https://doi.org/10.48550/arxiv.2202.01391" @default.
- W4288049772 hasPublicationYear "2022" @default.
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