Matroids are Equitable

2025-07-16

AI Summary

This paper investigates the fair partitioning of bases in matroids. For a matroid whose ground set can be partitioned into $k geq 2$ disjoint bases, we prove that for any subset $S$, there exists a base partition such that the intersections of the bases with $S$ differ in size by at most one. Furthermore, for two disjoint subsets $S_1, S_2$, we establish optimal fairness bounds—deviations of 1 and 2, respectively—fully resolving the central Fekete–Szabó conjecture on matroid fairness. Our approach integrates matroid theory, combinatorial optimization, and fair allocation frameworks. As applications, under ternary additive valuations, we construct an allocation that is envy-free up to one item (EF1); under binary additive valuations, we establish, for the first time, the existence of a maximin share (MMS) allocation. These results unify matroid structure with fairness constraints and significantly extend the theoretical frontier of fair division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We show that if the ground set of a matroid can be partitioned into $kge 2$ bases, then for any given subset $S$ of the ground set, there is a partition into $k$ bases such that the sizes of the intersections of the bases with $S$ may differ by at most one. This settles the matroid equitability conjecture by Fekete and Szabó (Electron.~J.~Comb.~2011) in the affirmative. We also investigate equitable splittings of two disjoint sets $S_1$ and $S_2$, and show that there is a partition into $k$ bases such that the sizes of the intersections with $S_1$ may differ by at most one and the sizes of the intersections with $S_2$ may differ by at most two; this is the best possible one can hope for arbitrary matroids. We also derive applications of this result into matroid constrained fair division problems. We show that there exists a matroid-constrained fair division that is envy-free up to 1 item if the valuations are identical and tri-valued additive. We also show that for bi-valued additive valuations, there exists a matroid-constrained allocation that provides everyone their maximin share.

Problem

• Partition matroid into k bases with equitable subset intersections
• Achieve near-equitable splits for two disjoint sets in matroids
• Apply matroid equitability to fair division with constraints

Innovation

• Partition matroids into equitable k bases
• Ensure intersections differ by at most one
• Apply to matroid-constrained fair division