[Paper Review] (Individual) Fairness for k-Clustering

Sepideh Mahabadi, Ali Vakilian|arXiv (Cornell University)|Jul 12, 2020

Facility Location and Emergency Management15 citations

TL;DR

This paper presents a local search algorithm for fair $k$-median and $k$-means clustering that achieves a bicriteria approximation: the clustering cost is within a constant factor of optimal fair clustering, and the fairness condition—where every point $x$ has a center within radius $r(x)$—is approximately satisfied. The approach ensures individual fairness by balancing cluster size and cost via radius-based fairness thresholds.

ABSTRACT

We give a local search based algorithm for $k$-median and $k$-means (and more generally for any $k$-clustering with $\ell_p$ norm cost function) from the perspective of individual fairness. More precisely, for a point $x$ in a point set $P$ of size $n$, let $r(x)$ be the minimum radius such that the ball of radius $r(x)$ centered at $x$ has at least $n/k$ points from $P$. Intuitively, if a set of $k$ random points are chosen from $P$ as centers, every point $x\in P$ expects to have a center within radius $r(x)$. An individually fair clustering provides such a guarantee for every point $x\in P$. This notion of fairness was introduced in [Jung et al., 2019] where they showed how to get an approximately feasible $k$-clustering with respect to this fairness condition. In this work, we show how to get a bicriteria approximation for fair $k$-clustering: The $k$-median ($k$-means) cost of our solution is within a constant factor of the cost of an optimal fair $k$-clustering, and our solution approximately satisfies the fairness condition (also within a constant factor). Further, we complement our theoretical bounds with empirical evaluation.

Motivation & Objective

• To address individual fairness in $k$-clustering by ensuring every point $x$ has a center within radius $r(x)$, where $r(x)$ is the minimal radius enclosing $n/k$ points.
• To develop a practical algorithm that approximates optimal fair $k$-median and $k$-means clustering while maintaining fairness guarantees.
• To provide theoretical bicriteria approximation bounds: constant-factor approximation in clustering cost and fairness violation.
• To empirically validate the trade-off between fairness and clustering cost in real-world settings.

Proposed method

• Define $r(x)$ as the smallest radius around point $x$ that contains at least $n/k$ points from the dataset $P$.
• Use a local search procedure to iteratively improve a $k$-clustering solution by swapping centers to reduce $\ell_p$ cost while tracking fairness violations.
• Maintain fairness by ensuring each point $x$ is within a constant factor of $r(x)$ from some center in the solution.
• Apply bicriteria analysis: the solution's cost is within a constant factor of the optimal fair clustering cost, and fairness is violated by at most a constant factor.
• Leverage the structure of $r(x)$ to bound the number of points per cluster and ensure balanced cluster sizes.

Experimental results

Research questions

• RQ1 Can a local search algorithm achieve a constant-factor approximation in cost while approximately satisfying individual fairness in $k$-clustering?
• RQ2 How does the fairness radius $r(x)$ relate to the optimal clustering cost under individual fairness constraints?
• RQ3 What is the trade-off between clustering cost and fairness violation in practice for $k$-median and $k$-means?
• RQ4 Can bicriteria approximation be achieved for general $\ell_p$ norm cost functions in fair $k$-clustering?

Key findings

• The proposed algorithm achieves a bicriteria approximation: the $k$-median and $k$-means cost is within a constant factor of the optimal fair clustering cost.
• The fairness condition is approximately satisfied, with each point $x$ having a center within a constant factor of $r(x)$.
• The algorithm maintains a constant approximation ratio for both cost and fairness, independent of $n$ or $k$.
• Empirical evaluation confirms the algorithm achieves a favorable trade-off between fairness and clustering cost on real datasets.