Improved Algorithms for Fair Matroid Submodular Maximization

TL;DR

This work advances Fair Matroid Monotone Submodular Maximization by introducing randomized and deterministic algorithms that achieve near-full fairness: for any fixed $\varepsilon>0$, the methods guarantee $\mathbb{E}[|S\cap V_c|] \ge (1-\varepsilon)\ell_c$ for all colors while maintaining matroid feasibility and upper-bound fairness. The randomized approach yields a constant-factor approximation to the objective, $\mathbb{E}[f(S)] \ge 0.499\cdot\varepsilon\cdot\operatorname{OPT}_{\mathrm{MatInt}}$, and strengthens tail bounds on solution size and total fairness violation; a deterministic variant for matroid intersections provides comparable guarantees. The authors validate their methods on graph coverage, clustering, and recommender-system datasets, demonstrating favorable utility alongside tunable fairness via $\varepsilon$. Overall, the paper offers principled, scalable tools to balance representation fairness with submodular utility in constrained selection problems with real-world societal relevance.

Abstract

Submodular maximization subject to matroid constraints is a central problem with many applications in machine learning. As algorithms are increasingly used in decision-making over datapoints with sensitive attributes such as gender or race, it is becoming crucial to enforce fairness to avoid bias and discrimination. Recent work has addressed the challenge of developing efficient approximation algorithms for fair matroid submodular maximization. However, the best algorithms known so far are only guaranteed to satisfy a relaxed version of the fairness constraints that loses a factor 2, i.e., the problem may ask for $\ell$ elements with a given attribute, but the algorithm is only guaranteed to find $\lfloor \ell/2 \rfloor$. In particular, there is no provable guarantee when $\ell=1$, which corresponds to a key special case of perfect matching constraints. In this work, we achieve a new trade-off via an algorithm that gets arbitrarily close to full fairness. Namely, for any constant $\varepsilon>0$, we give a constant-factor approximation to fair monotone matroid submodular maximization that in expectation loses only a factor $(1-\varepsilon)$ in the lower-bound fairness constraint. Our empirical evaluation on a standard suite of real-world datasets -- including clustering, recommendation, and coverage tasks -- demonstrates the practical effectiveness of our methods.

Figures (2)

• Figure 1: Our experimental results. Each row corresponds to one experiment; the left plot shows the objective value of each algorithm for a range of solution scale factors $r$, and the right plot shows fairness violations. For randomized algorithms we report averages, with error bars that correspond to sample standard deviation.
• Figure 2: For each experiment and algorithm we take the average objective value and fairness violation over all $r$-values, and plot this as a single point. For randomized algorithms, the colored rectangles correspond to standard deviations. The dashed line corresponds to the Pareto frontier of the trade-off between objective value and fairness violation.

Theorems & Definitions (16)

• Theorem 1.1: Two-pass algorithm of el2023fairness
• Theorem 1.2: el2024
• Theorem 1.3: informal version of \ref{['thm:main-randomized']}
• Theorem 1.4: informal version of \ref{['thm:main-deterministic']}
• Definition 2.2
• Lemma 2.3: Schrijver, Corollary 39.12a
• Lemma 2.4: Schrijver, Corollary 39.13
• Lemma 2.5: el2023fairness, Appendix C
• Theorem 2.6: CalinescuCPV11
• Theorem 2.7: Lee2010