A Polynomial-Time Approximation for Pairwise Fair $k$-Median Clustering

TL;DR

This work tackles Pairwise Fair $k$-Median clustering with $\\ell$ groups and a uniform balance parameter $t$, aiming to minimize the $k$-median cost while strictly satisfying fairness constraints. The authors present the first true polynomial-time $O(k^2 \cdot \ell \cdot t)$-approximation by leveraging centers from a standard $k$-median approximation, a linear-programming relaxation to obtain a near-fair assignment, and a careful rounding plus a controlled reassignment phase. They establish hardness results showing the problem is as hard as Soft Uniform Capacitated $k$-Median for disjoint groups (unless poly-time breakthroughs occur) and NP-hard to approximate within $n^{1-\varepsilon}$ for overlapping groups, via reductions from capacitated problems and hypergraph 2-colorability. Experiments on real datasets illustrate practical performance, revealing the cost of enforcing fairness grows with the number of clusters but remains favorable compared to the theoretical worst-case bounds, with runtime largely driven by the initial $k$-median step.

Abstract

In this work, we study pairwise fair clustering with $\ell \ge 2$ groups, where for every cluster $C$ and every group $i \in [\ell]$, the number of points in $C$ from group $i$ must be at most $t$ times the number of points in $C$ from any other group $j \in [\ell]$, for a given integer $t$. To the best of our knowledge, only bi-criteria approximation and exponential-time algorithms follow for this problem from the prior work on fair clustering problems when $\ell > 2$. In our work, focusing on the $\ell > 2$ case, we design the first polynomial-time $O(k^2\cdot \ell \cdot t)$-approximation for this problem with $k$-median cost that does not violate the fairness constraints. We complement our algorithmic result by providing hardness of approximation results, which show that our problem even when $\ell=2$ is almost as hard as the popular uniform capacitated $k$-median, for which no polynomial-time algorithm with an approximation factor of $o(\log k)$ is known.

Figures (3)

• Figure 1: (a) A fair representational clustering for three groups red (r, disk), blue (b, square), and green (g, cross). Here $\alpha_r=1/3,\beta_r=2/3,\alpha_b=1/6,\beta_b=1/3,\alpha_g=1/6,\beta_g=1/3$. $C_1,C_2,C_3$ are the clusters. (b) A pairwise fair clustering with $t=2$ for the same dataset. The clustering in (a) is not pairwise fair for $t=2$, as $C_1$ contains 4 red (disk) points, but only 1 blue (square).
• Figure 2: Cost comparison across multiple datasets with varying target numbers of clusters ($k$).
• Figure 3: Runtime comparison across multiple datasets with varying numbers of clusters ($k$).

Theorems & Definitions (24)

• Definition 1: Fair Representational Clustering
• Definition 2: Pairwise Fair Clustering
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• proof
• Lemma 3