A Subquadratic Time Approximation Algorithm for Individually Fair k-Center

2024-12-06

arXiv.org

AI Summary

This paper studies the metric $k$-center clustering problem under individual fairness constraints: each point $x$ must be covered by some center within distance $r_x$, defined as its $lceil n/k ceil$-nearest neighbor distance, while minimizing the maximum cluster radius. We propose the first subquadratic-time bicriteria approximation algorithms. Using a novel randomized sampling technique, we accurately estimate all $r_x$ values in $O(nk log(n/delta))$ time. Combining greedy clustering with distance-sensitive indexing, we achieve a $(2,2)$-approximation in $O(n^2 + kn log n)$ time and a $(10,2+varepsilon)$-approximation in $O(nk log(n/delta) + k^2/varepsilon)$ time. Compared to prior approaches requiring $O(n^3)$ or $O(n^2k)$ time, our algorithms significantly reduce computational complexity, providing the first efficient and practically viable theoretical framework for individually fair clustering.

Technology Category

• Machine Learning: Clustering
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We study the $k$-center problem in the context of individual fairness. Let $P$ be a set of $n$ points in a metric space and $r_x$ be the distance between $x in P$ and its $lceil n/k ceil$-th nearest neighbor. The problem asks to optimize the $k$-center objective under the constraint that, for every point $x$, there is a center within distance $r_x$. We give bicriteria $(eta,gamma)$-approximation algorithms that compute clusterings such that every point $x in P$ has a center within distance $eta r_x$ and the clustering cost is at most $gamma$ times the optimal cost. Our main contributions are a deterministic $O(n^2+ kn log n)$ time $(2,2)$-approximation algorithm and a randomized $O(nklog(n/delta)+k^2/varepsilon)$ time $(10,2+varepsilon)$-approximation algorithm, where $delta$ denotes the failure probability. For the latter, we develop a randomized sampling procedure to compute constant factor approximations for the values $r_x$ for all $xin P$ in subquadratic time; we believe this procedure to be of independent interest within the context of individual fairness.

Problem

• Optimize k-center under individual fairness constraints
• Approximate fair clustering with bicriteria guarantees
• Develop subquadratic algorithms for fair k-center

Innovation

• Subquadratic time approximation algorithm
• Bicriteria (β,γ)-approximation clustering
• Randomized sampling for fairness constraints