On the Hardness of Approximation of the Fair k-Center Problem

2026-02-18

AI Summary

This study investigates the approximability of the fair k-center problem, which seeks to minimize the maximum distance from any point to its nearest center under the constraint that a specified number of centers must be selected from each group. Leveraging computational complexity theory, reduction techniques, and metric space analysis, the work establishes—for the first time—that achieving a (3−ε)-approximation is NP-hard for any ε > 0, even in highly restricted settings such as when there are only two disjoint groups or when exactly one center must be chosen from each group. This result establishes a tight theoretical lower bound of 3 on the approximation factor, demonstrating that existing 3-approximation algorithms are optimal under various constrained formulations of the problem.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Game Theory and Economic Paradigms: Fair Division
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

In this work, we study the hardness of approximation of the fair $k$-center problem. Here the data points are partitioned into groups and the task is to choose a prescribed number of data points from each group, called centers, while minimizing the maximum distance from any point to its closest center. Although a polynomial-time $3$-approximation is known for this problem in general metrics, it has remained open whether this approximation guarantee is tight or could be further improved, especially since the unconstrained $k$-center problem admits a polynomial-time factor-$2$ approximation. We resolve this open question by proving that, for every $ε>0$, achieving a $(3-ε)$-approximation is NP-hard, assuming $\text{P} \neq \text{NP}$. Our inapproximability results hold even when only two disjoint groups are present and at least one center must be chosen from each group. Further, it extends to the canonical one-per-group setting with $k$-groups (for arbitrary $k$), where exactly one center must be selected from each group. Consequently, the factor-$3$ barrier for fair $k$-center in general metric spaces is inherent, and existing $3$-approximation algorithms are optimal up to lower-order terms even in these restricted regimes. This result stands in sharp contrast to the $k$-supplier formulation, where both the unconstrained and fair variants admit factor-$3$ approximation in polynomial time.

Problem

• fair k-center
• hardness of approximation
• NP-hard
• approximation ratio
• clustering

Innovation

• fair k-center
• hardness of approximation
• NP-hardness
• metric spaces
• approximation algorithms