[Paper Review] Approximation Algorithms for Fair Range Clustering

Sèdjro S. Hotegni, Sepideh Mahabadi+1 more|arXiv (Cornell University)|Jun 11, 2023

Facility Location and Emergency Management4 citations

TL;DR

This paper presents the first constant-factor approximation algorithms for fair range ℓp-clustering, a generalization of fair k-center, k-median, and k-means clustering. By leveraging a structured linear program and a novel network flow construction, the authors achieve an O(1)-approximation for all p ∈ [1, ∞), resolving an open problem in fair clustering with relaxed group representation constraints.

ABSTRACT

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick $k$ centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of $n$ points in a metric space $(P,d)$ where each point belongs to one of the $\ell$ different demographics (i.e., $P = P_1 \uplus P_2 \uplus \cdots \uplus P_\ell$) and a set of $\ell$ intervals $[α_1, β_1], \cdots, [α_\ell, β_\ell]$ on desired number of centers from each group, the goal is to pick a set of $k$ centers $C$ with minimum $\ell_p$-clustering cost (i.e., $(\sum_{v\in P} d(v,C)^p)^{1/p}$) such that for each group $i\in \ell$, $|C\cap P_i| \in [α_i, β_i]$. In particular, the fair range $\ell_p$-clustering captures fair range $k$-center, $k$-median and $k$-means as its special cases. In this work, we provide efficient constant factor approximation algorithms for fair range $\ell_p$-clustering for all values of $p\in [1,\infty)$.

Motivation & Objective

• To address fairness in centroid-based clustering by ensuring minimal and maximal representation of each demographic group in the selected centers.
• To generalize the strict fair clustering model (fixed number of centers per group) to a more flexible 'fair range' model with lower and upper bounds on group representation.
• To design efficient approximation algorithms for the fair range ℓp-clustering problem with multiplicative guarantees for all p ∈ [1, ∞).
• To close the gap in approximability for fair range clustering with ℓp objectives, which remained open despite progress on special cases.
• To provide a practical and theoretically sound framework for fair data summarization in real-world applications like image search and resource allocation.

Proposed method

• Formulate a structured linear program (StructuredLP) for a sparse instance of the fair range clustering problem to reduce complexity.
• Construct a layered network flow instance with source, facility, group, and sink layers to model center selection and group constraints.
• Use a half-integral optimal solution of StructuredLP to guide flow construction, ensuring feasibility and integral center selection.
• Route flow from source to sink such that each facility receives flow proportional to its selection probability, respecting group bounds and total center count k.
• Prove that the resulting center set satisfies all fairness constraints (αi ≤ |C ∩ Pi| ≤ βi) and achieves bounded clustering cost.
• Establish approximation ratio by relating the cost of the constructed solution to the optimal solution of the linear program, yielding O(1) approximation after raising to power 1/p.

Experimental results

Research questions

• RQ1 Can we design efficient constant-factor approximation algorithms for fair range ℓp-clustering with general p ∈ [1, ∞)?
• RQ2 How can we relax the strict requirement of fixed center counts per group to improve solution quality while preserving fairness?
• RQ3 What is the approximability of fair range clustering under ℓp objectives, particularly for k-median and k-means?
• RQ4 Can we achieve multiplicative approximation guarantees using LP relaxation and network flow techniques in this constrained setting?
• RQ5 Is there a way to transform a fractional LP solution into an integral center set that satisfies group bounds and maintains low clustering cost?

Key findings

• The paper achieves an O(1)-approximation for fair range ℓp-clustering for all p ∈ [1, ∞), providing the first pure multiplicative approximation for this problem.
• The approximation factor is bounded by (9/2)^p times the optimal LP cost, which translates to a constant factor after adjusting for the ℓp norm.
• A novel network flow construction ensures that all fairness constraints (αi ≤ |C ∩ Pi| ≤ βi) are satisfied while maintaining total center count k.
• The method relies on a sparse instance reduction and a half-integral LP solution to enable efficient flow routing and integral center selection.
• The approach generalizes to k-center, k-median, and k-means as special cases of ℓp-clustering with p = ∞, 1, and 2, respectively.
• The framework is the first to provide a constant-factor approximation for fair range clustering with general ℓp objectives, resolving an open problem in the field.