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

TL;DR

This paper proves that achieving better than a 3-approximation for the fair k-center problem is NP-hard, establishing the optimality of existing algorithms and highlighting a fundamental complexity barrier.

Contribution

It establishes the NP-hardness of approximating fair k-center within any factor less than 3, confirming the optimality of current algorithms and contrasting with related problems.

Findings

No polynomial-time algorithm can approximate fair k-center within (3 - epsilon) for any epsilon > 0.

The hardness result holds even with only two groups and one center per group.

Existing 3-approximation algorithms are essentially optimal under standard complexity assumptions.

Abstract

In this work, we study the hardness of approximation of the fair $k$-center problem. In this problem, we are given a set of data points in a metric space that is partitioned into groups and the task is to choose a subset of $k$-data points, called centers, such that a prescribed number of data points from each group are chosen while minimizing the maximum distance from any point to its closest center. Although a polynomial-time $3$-approximation is known for fair $k$-center in general metrics, it has remained open whether this approximation guarantee is tight or could be further improved, especially since the classical unconstrained $k$-center problem admits a polynomial-time factor-$2$ approximation. We resolve this open question by proving that, assuming $\mathsf{P} \neq \mathsf{NP}$, for any $\epsilon>0$, no polynomial-time algorithm can approximate fair $k$-center to $(3-\epsilon)$-factor. 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.