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

Suhas Thejaswi

TL;DR

It is proved that, for every $\epsilon>0$, achieving a $(3-\epsilon)$-approximation is NP-hard, assuming $\text{P} \neq \text{NP}$.

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.

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

TL;DR

It is proved that, for every , achieving a -approximation is NP-hard, assuming .

Abstract

In this work, we study the hardness of approximation of the fair -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 -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 -center problem admits a polynomial-time factor- approximation. We resolve this open question by proving that, for every , achieving a -approximation is NP-hard, assuming . 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 -groups (for arbitrary ), where exactly one center must be selected from each group. Consequently, the factor- barrier for fair -center in general metric spaces is inherent, and existing -approximation algorithms are optimal up to lower-order terms even in these restricted regimes. This result stands in sharp contrast to the -supplier formulation, where both the unconstrained and fair variants admit factor- approximation in polynomial time.
Paper Structure (7 sections, 2 theorems, 14 equations)

This paper contains 7 sections, 2 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Problem definitions
  3. Hardness of approximation
  4. Two groups regime
  5. One-per-group regime
  6. Further related work
  7. Conclusions

Key Result

Theorem 1

Assuming $\mathsf{P}\xspace \neq \mathsf{NP}\xspace$, for any $\epsilon > 0$, there exists no polynomial-time $(3-\epsilon)$-approximation algorithm for the fair $k$-center problem even when there are only two groups and at least one center must be selected from each group.

Theorems & Definitions (24)

  • Definition 1: The fair $k$-center problem
  • Definition 2: The one-per-group fair $k$-center problem
  • Definition 3: The $k$-center problem with forbidden centers
  • Theorem 1
  • proof
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Claim 3
  • ...and 14 more