[Paper Review] Improved Approximation Algorithms for Individually Fair Clustering

Ali Vakilian, Mustafa Yalçiner|arXiv (Cornell University)|Jun 26, 2021

Facility Location and Emergency Management35 references4 citations

TL;DR

This paper presents improved bicriteria approximation algorithms for individually fair clustering under ℓp-norm cost, achieving a (16^p + ε, 3)-approximation for general p > 1 and (7.081 + ε, 3) for k-median (p=1). The approach extends facility location with matroid constraints and reduces fair clustering to a group fairness problem, significantly improving over prior (p^O(p), 7) guarantees.

ABSTRACT

We consider the $k$-clustering problem with $\ell_p$-norm cost, which includes $k$-median, $k$-means and $k$-center, under an individual notion of fairness proposed by Jung et al. [2020]: given a set of points $P$ of size $n$, a set of $k$ centers induces a fair clustering if every point in $P$ has a center among its $n/k$ closest neighbors. Mahabadi and Vakilian [2020] presented a $(p^{O(p)},7)$-bicriteria approximation for fair clustering with $\ell_p$-norm cost: every point finds a center within distance at most $7$ times its distance to its $(n/k)$-th closest neighbor and the $\ell_p$-norm cost of the solution is at most $p^{O(p)}$ times the cost of an optimal fair solution. In this work, for any $\varepsilon>0$, we present an improved $(16^p +\varepsilon,3)$-bicriteria for this problem. Moreover, for $p=1$ ($k$-median) and $p=\infty$ ($k$-center), we present improved cost-approximation factors $7.081+\varepsilon$ and $3+\varepsilon$ respectively. To achieve our guarantees, we extend the framework of [Charikar et al., 2002, Swamy, 2016] and devise a $16^p$-approximation algorithm for the facility location with $\ell_p$-norm cost under matroid constraint which might be of an independent interest. Besides, our approach suggests a reduction from our individually fair clustering to a clustering with a group fairness requirement proposed by Kleindessner et al. [2019], which is essentially the median matroid problem [Krishnaswamy et al., 2011].

Motivation & Objective

• To address the gap in efficient approximation algorithms for individually fair clustering under ℓp-norm cost functions.
• To improve upon the prior (p^O(p), 7)-bicriteria approximation for fair k-clustering with ℓp costs.
• To achieve better fairness and cost approximation factors, especially for k-median and k-center variants.
• To develop a novel facility location algorithm under matroid constraints with ℓp-cost, of independent interest.
• To establish a reduction from individually fair clustering to group-fair clustering via the median matroid problem.

Proposed method

• Extends the facility location framework of Charikar et al. (2002) and Swamy (2016) to handle ℓp-cost under matroid constraints.
• Introduces a 16^p-approximation algorithm for matroid-constrained facility location with ℓp-cost, which is central to the main result.
• Employs a critical region decomposition of the point set to group points by their (n/k)-nearest neighbor distances.
• Constructs a transformed instance P' with copies of points in critical regions and applies a partition matroid constraint to enforce center distribution.
• Uses a center-partitioning algorithm on the transformed instance to compute a solution that preserves fairness and cost bounds.
• Reduces the individually fair clustering problem to a group-fair clustering problem via the median matroid problem, enabling use of existing approximation techniques.

Experimental results

Research questions

• RQ1 Can we achieve better fairness and cost approximation factors than the (p^O(p), 7) guarantee for individually fair ℓp-clustering?
• RQ2 Is it possible to design a constant-factor cost approximation for k-median and k-center under individual fairness?
• RQ3 Can the matroid-constrained facility location problem with ℓp-cost be approximated within a factor of 16^p?
• RQ4 Does a reduction from individually fair clustering to group-fair clustering via the median matroid problem yield improved approximations?
• RQ5 Can the bicriteria approximation factor for fairness be reduced from 7 to 3 while maintaining constant cost approximation?

Key findings

• The paper achieves a (16^p + ε, 3)-bicriteria approximation for α-fair k-clustering with ℓp-cost, improving over the prior (p^O(p), 7) guarantee.
• For k-median (p=1), the algorithm achieves a (7.081 + ε, 3)-bicriteria approximation, significantly improving on the prior (84, 7) result.
• For k-center (p=∞), the algorithm achieves a (3 + ε, 3)-bicriteria approximation, improving on the prior (O(log n), 7) bound.
• A novel 16^p-approximation algorithm is developed for facility location with ℓp-cost under matroid constraints, which may be of independent interest.
• The reduction to group fairness via the median matroid problem enables leveraging existing approximation techniques for improved fairness and cost bounds.
• The cost approximation factor is bounded by β + ε, where β is the approximation ratio of the underlying facility location algorithm, and ε allows for arbitrarily small additive error.