Approximate the individually fair k-center with outliers

TL;DR

This paper introduces the individually fair k-center with outliers problem, proposing a 4-approximation algorithm and an improved practical algorithm for selecting centers and outliers in metric spaces while ensuring fairness.

Contribution

It formulates the new IFkCO problem with fairness constraints and provides the first approximation algorithm, along with an improved practical approach.

Findings

Presented a 4-approximation algorithm for IFkCO.

Developed an improved algorithm for practical applications.

Analyzed the fairness ratio minimization in outlier-inclusive clustering.

Abstract

In this paper, we propose and investigate the individually fair $k$-center with outliers (IF$k$CO). In the IF$k$CO, we are given an $n$-sized vertex set in a metric space, as well as integers $k$ and $q$. At most $k$ vertices can be selected as the centers and at most $q$ vertices can be selected as the outliers. The centers are selected to serve all the not-an-outlier (i.e., served) vertices. The so-called individual fairness constraint restricts that every served vertex must have a selected center not too far way. More precisely, it is supposed that there exists at least one center among its $\lceil (n-q) / k \rceil$ closest neighbors for every served vertex. Because every center serves $(n-q) / k$ vertices on the average. The objective is to select centers and outliers, assign every served vertex to some center, so as to minimize the maximum fairness ratio over all served vertices, where the fairness ratio of a vertex is defined as the ratio between its distance with the assigned center and its distance with a $\lceil (n - q )/k \rceil_{\rm th}$ closest neighbor. As our main contribution, a 4-approximation algorithm is presented, based on which we develop an improved algorithm from a practical perspective.