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Fair Clustering for Data Summarization: Improved Approximation Algorithms and Complexity Insights

Ameet Gadekar, Aristides Gionis, Suhas Thejaswi

TL;DR

This work focuses on fair data summarization modeled as the fair k-supplier problem, where data consists of several groups, and a minimum number of centers must be selected from each group while minimizing the k-supplier objective.

Abstract

Data summarization tasks are often modeled as $k$-clustering problems, where the goal is to choose $k$ data points, called cluster centers, that best represent the dataset by minimizing a clustering objective. A popular objective is to minimize the maximum distance between any data point and its nearest center, which is formalized as the $k$-center problem. While in some applications all data points can be chosen as centers, in the general setting, centers must be chosen from a predefined subset of points, referred as facilities or suppliers; this is known as the $k$-supplier problem. In this work, we focus on fair data summarization modeled as the fair $k$-supplier problem, where data consists of several groups, and a minimum number of centers must be selected from each group while minimizing the $k$-supplier objective. The groups can be disjoint or overlapping, leading to two distinct problem variants each with different computational complexity. We present $3$-approximation algorithms for both variants, improving the previously known factor of $5$. For disjoint groups, our algorithm runs in polynomial time, while for overlapping groups, we present a fixed-parameter tractable algorithm, where the exponential runtime depends only on the number of groups and centers. We show that these approximation factors match the theoretical lower bounds, assuming standard complexity theory conjectures. Finally, using an open-source implementation, we demonstrate the scalability of our algorithms on large synthetic datasets and assess the price of fairness on real-world data, comparing solution quality with and without fairness constraints.

Fair Clustering for Data Summarization: Improved Approximation Algorithms and Complexity Insights

TL;DR

This work focuses on fair data summarization modeled as the fair k-supplier problem, where data consists of several groups, and a minimum number of centers must be selected from each group while minimizing the k-supplier objective.

Abstract

Data summarization tasks are often modeled as -clustering problems, where the goal is to choose data points, called cluster centers, that best represent the dataset by minimizing a clustering objective. A popular objective is to minimize the maximum distance between any data point and its nearest center, which is formalized as the -center problem. While in some applications all data points can be chosen as centers, in the general setting, centers must be chosen from a predefined subset of points, referred as facilities or suppliers; this is known as the -supplier problem. In this work, we focus on fair data summarization modeled as the fair -supplier problem, where data consists of several groups, and a minimum number of centers must be selected from each group while minimizing the -supplier objective. The groups can be disjoint or overlapping, leading to two distinct problem variants each with different computational complexity. We present -approximation algorithms for both variants, improving the previously known factor of . For disjoint groups, our algorithm runs in polynomial time, while for overlapping groups, we present a fixed-parameter tractable algorithm, where the exponential runtime depends only on the number of groups and centers. We show that these approximation factors match the theoretical lower bounds, assuming standard complexity theory conjectures. Finally, using an open-source implementation, we demonstrate the scalability of our algorithms on large synthetic datasets and assess the price of fairness on real-world data, comparing solution quality with and without fairness constraints.

Paper Structure

This paper contains 8 sections, 3 theorems, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Approximation algorithms
  4. Experiments
  5. Related work
  6. Conclusions, limitations and open problems
  7. Additional experimental results
  8. Proof of Theorem \ref{['thm:fairksupplier:3apx']}

Key Result

Theorem 3.1

There is a $3$-approximation algorithm for the problem $\text{\sc Fair-\-}k\text{\sc-Sup}$-$\varnothing$ with runtime $\mathsf{O}\xspace((kn+k^2\sqrt{k}) \log n \log k)$. Furthermore, assuming $\mathsf{P}\xspace \neq \mathsf{NP}\xspace$, no polynomial-time algorithm achieves $(3-\epsilon)$-approxima

Figures (3)

  • Figure 1: Scalability of the $3$-approximation algorithm (Algorithm \ref{['alg:3apx']}) and the $5$-approximation by chen2024approximation for $\text{\sc Fair-\-}k\text{\sc-Sup-}\varnothing$ with $t=5$ disjoint groups and fairness requirements $\vec{\alpha} = [\frac{k}{t}]^{t}$.
  • Figure 2: Scalability of the $3$-approximation algorithm (Algorithm \ref{['alg:gen3apx']}) and the $5$-approximation for $\text{\sc Fair-\-}k\text{\sc-Sup}$ with $t=4$ intersecting groups, where the requirements are uniform across groups with $\vec{\alpha} = [2 \cdot \frac{k}{t}]^t$.
  • Figure 3: Scalability of the $3$-approximation algorithm (Algorithm \ref{['alg:3apx']}) with respect to number of clients $n_c$ and number of facilities $n_f$ for $\text{\sc Fair-\-}k\text{\sc-Sup-}\varnothing$ with $t=5$ disjoint groups and fairness requirements $\vec{\alpha} = [\frac{k}{t}]^{t}$.

Theorems & Definitions (5)

  • Definition 2.1: The fair $k$-supplier problem
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Lemma B.1