Facility Location and $k$-Median with Fair Outliers

TL;DR

This paper introduces fair outlier constraints into Facility Location and $k$-Median clustering problems, providing approximation algorithms that respect group fairness while maintaining low costs, supported by empirical validation.

Contribution

It develops bicriteria approximation algorithms for fair outlier variants of Facility Location and $k$-Median, improving previous bounds and avoiding dependence on $k$ for outlier violations.

Findings

Fair outlier constraints can be incorporated with minimal cost increase.

The algorithms achieve provable approximation guarantees with group fairness.

Empirical results show practical effectiveness and small LP integrality gaps.

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.