Polynomial-Time Constant-Approximation for Fair Sum-of-Radii Clustering

2025-04-20

AI Summary

This paper studies the Sum-of-Radii clustering problem in metric spaces under proportional fairness constraints: given red and blue point sets, each cluster must satisfy $1/t leq r/b leq t$ (where $r$ and $b$ are the numbers of red and blue points), use at most $k$ clusters, and minimize the sum of cluster radii. We present the first polynomial-time $O(1)$-approximation algorithm. Its core innovation is a unified analytical framework based on iterative cluster merging, which integrates the approximation complexity of Sum-of-Radii with that of $k$-center and related objectives into a single theoretical paradigm. The algorithm runs in $mathrm{poly}(n,k)$ time, naturally extends to $ell geq 2$ colors (yielding an $O(1)$-approximation when $t=1$), and—critically—yields the first polynomial-time $O(1)$-approximation for Euclidean Sum-of-Radii clustering under proportional fairness. This work significantly advances both the theoretical foundations and algorithmic landscape of fair clustering.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Clustering

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

In a seminal work, Chierichetti et al. introduced the $(t,k)$-fair clustering problem: Given a set of red points and a set of blue points in a metric space, a clustering is called fair if the number of red points in each cluster is at most $t$ times and at least $1/t$ times the number of blue points in that cluster. The goal is to compute a fair clustering with at most $k$ clusters that optimizes certain objective function. Considering this problem, they designed a polynomial-time $O(1)$- and $O(t)$-approximation for the $k$-center and the $k$-median objective, respectively. Recently, Carta et al. studied this problem with the sum-of-radii objective and obtained a $(6+epsilon)$-approximation with running time $O((klog_{1+epsilon}(k/epsilon))^kn^{O(1)})$, i.e., fixed-parameter tractable in $k$. Here $n$ is the input size. In this work, we design the first polynomial-time $O(1)$-approximation for $(t,k)$-fair clustering with the sum-of-radii objective, improving the result of Carta et al. Our result places sum-of-radii in the same group of objectives as $k$-center, that admit polynomial-time $O(1)$-approximations. This result also implies a polynomial-time $O(1)$-approximation for the Euclidean version of the problem, for which an $f(k)cdot n^{O(1)}$-time $(1+epsilon)$-approximation was known due to Drexler et al.. Here $f$ is an exponential function of $k$. We are also able to extend our result to any arbitrary $ellge 2$ number of colors when $t=1$. This matches known results for the $k$-center and $k$-median objectives in this case. The significant disparity of sum-of-radii compared to $k$-center and $k$-median presents several complex challenges, all of which we successfully overcome in our work. Our main contribution is a novel cluster-merging-based analysis technique for sum-of-radii that helps us achieve the constant-approximation bounds.

Problem

• Develop polynomial-time O(1)-approximation for fair sum-of-radii clustering
• Improve existing results for (t,k)-fair clustering with sum-of-radii objective
• Extend solution to arbitrary number of colors when t=1

Innovation

• Polynomial-time O(1)-approximation for fair clustering
• Novel cluster-merging-based analysis technique
• Extends to arbitrary number of colors when t=1