Facility Location and $k$-Median with Fair Outliers

Abstract: Classical clustering problems such as \emph{Facility Location} and \emph{$k$-Median} aim to efficiently serve a set of clients from a subset of facilities -- minimizing the total cost of facility openings and client assignments in Facility Location, and minimizing assignment (service) cost under a facility count constraint in $k$-Median. These problems are highly sensitive to outliers, and therefore researchers have studied variants that allow excluding a small number of clients as outliers to reduce cost. However, in many real-world settings, clients belong to different demographic or functional groups, and unconstrained outlier removal can disproportionately exclude certain groups, raising fairness concerns. We study \emph{Facility Location with Fair Outliers}, where each group is allowed a specified number of outliers, and the objective is to minimize total cost while respecting group-wise fairness constraints. We present a bicriteria approximation with a $O(1/\epsilon)$ approximation factor and $(1+ 2\epsilon)$ factor violation in outliers per group. For \emph{$k$-Median with Fair Outliers}, we design a bicriteria approximation with a $4(1+\omega/\epsilon)$ approximation factor and $(\omega + \epsilon)$ violation in outliers per group improving on prior work by avoiding dependence on $k$ in outlier violations. We also prove that the problems are W[1]-hard parameterized by $\omega$, assuming the Exponential Time Hypothesis. We complement our algorithmic contributions with a detailed empirical analysis, demonstrating that fairness can be achieved with negligible increase in cost and that the integrality gap of the standard LP is small in practice.