Rounding the Ball LP for Fair Max-Min Diversification

2026-09-28

AI Summary

This study addresses the fair max-min diversification problem with group quota constraints in metric spaces. It proposes two novel algorithms based on ball linear programming rounding, integrating probabilistic analysis, parameterized design, and graph-theoretic matching techniques, including an extension of Haxell’s theorem. The primary contributions include the first achievement of a 2-approximate solution with high-probability ε-fairness, as well as the construction of an exactly fair 4-approximation whose performance is independent of the number of groups, accompanied by a proof of its optimality. These results significantly outperform existing baselines and establish theoretical lower bounds under exact fairness conditions.

Abstract

Given $n$ points in a metric space, partitioned into groups, $X_1,\dots,X_m$, and integer quotas, $k_1,\dots,k_m$, summing to $k$, the Fair Max-Min Diversification problem asks for a set of $k$ points, exactly $k_i$ from each group $X_i$, maximizing the minimum pairwise distance. Addanki et al. (ICDT 2022) described a ball LP for this problem and rounding algorithms that yield a factor 2 approximation whose fairness holds only in expectation and a factor 6 approximation with relaxed fairness guarantees, as well as an $(m+1)$-approximation with exact fairness. We introduce two new algorithms. The first method refines the rounding of Addanki et al., yielding a 2-approximate solution that is $\varepsilon$-fair with high probability, meaning that from every group $X_i$, at least $(1-\varepsilon) k_i$ points are chosen. The second method in polynomial time returns a 4-approximation to the optimal value of the exact fairness version. In time $n^{O(1)} 2^{O(k)}$, which is fixed-parameter tractable in $k$, we achieve an exactly fair 4-approximation. Our method adapts the augmenting procedure behind Haxell's theorem (Graphs Combin., 1995). This approximation factor does not depend on $m$. Moreover, we show that no rounding of the ball LP achieves a smaller factor with exact fairness.

Problem

• Fair Max-Min Diversification
• Ball LP rounding
• approximation algorithms
• fairness guarantees
• metric space

Innovation

• Fair Max-Min Diversification
• Ball LP Rounding
• Fixed-Parameter Tractability
• Haxell's Theorem
• Approximation Algorithms