Matroids are Equitable

TL;DR

The paper proves that any matroid whose ground set is a union of k bases can be partitioned into k bases so that each base intersects any fixed subset S of the ground set by nearly equal amounts, generalizing the matroid equitability conjecture and providing a polynomial-time algorithm. The authors develop a structural exchange framework using an exchange graph to locate exchangeable sets, enabling iterative rebalancing of base intersections. They extend the results to two subsets, establishing near-equitability bounds that are tight in general, and translate these combinatorial insights into fair-division applications, yielding EF1 allocations under identical binary or tri-valued additive valuations and MMS allocations under bi-valued valuations. The work connects to and strengthens several conjectures in matroid theory (e.g., White, Gabow) and offers constructive tools for matroid-constrained fair division problems with practical implications for equitable resource allocation. Overall, it provides a unifying, algorithmic approach to equitable partitioning and its application to fairness notions in discrete allocations.

Abstract

We show that if the ground set of a matroid can be partitioned into $k\ge 2$ 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 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 one 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.

Figures (2)

• Figure 1: The elements in $S$ are colored red. The blue set illustrates $K$.
• Figure 2: Solid edges correspond to the elements in $B_1$ and dashed edges to the elements in $B_2$. If $(C_1,C_2)$ equally divide $S_1 = \{(1,2), (3,4)\}$ and $S_2 = \{(1,3), (2,4)\}$, by symmetry we can assume $\{(1,2),(1,3)\} \subset C_1$ and $\{(2,4),(3,4)\} \subset C_2$. The set that includes $(2,3)$ has a cycle and therefore is not independent.

Theorems & Definitions (44)

• Theorem 1.1
• Corollary 1.2: of Theorem \ref{['thm:two-equitable']}
• Definition 1.3
• Theorem 1.4
• Conjecture 1.5
• Definition 1.6
• Definition 1.7: EF1
• Corollary 1.8: of Theorem \ref{['thm:main-equitable']}
• Theorem 1.9
• Definition 1.10