Generalizing the Multiple Exchange Property for Matroid Bases
Taihei Oki, Tamás Schwarcz
TL;DR
This paper introduces a wide-reaching generalization of the multiple exchange property for matroid bases, proving that for any A,B bases and X⊆A∖B, Y⊆B∖A there exist U⊆A∖B and V⊆B∖A with X⊆U, Y⊆V, |U|=|V|≤r(X+Y), and both A−U+V and B+U−V are bases. The authors develop an algorithmic proof via weighted matroid intersection and Frank's weight splitting, yielding a polynomial-time method to find such U,V and establishing robustness results for local-search algorithms. They further generalize Grassmann–Plücker identities to characteristic-zero representable matroids (and extend to valuated matroids), providing a framework that subsumes and strengthens several prior exchange properties (Greene, Kung) and yields corollaries related to base orderability and Gabow-type conjectures. The work also offers randomized algorithms for finding exchangeable sets, and applications to equitability in fair division under matroid constraints, as well as a series of extensions to valuated matroids and potential future directions in serial exchange and Gabow-like decompositions. Overall, the paper advances fundamental understanding of matroid bases, links combinatorial optimization with algebraic identities, and demonstrates algorithmic and economic implications of generalized exchange properties.
Abstract
The multiple exchange property for matroid bases states that for any bases $A$ and $B$ of a matroid and any subset $X\subseteq A\setminus B$, there exists a subset $Y\subseteq B\setminus A$ such that both $A-X+Y$ and $B+X-Y$ are bases. This classical result has not only found applications in matroid theory, but also in the analysis and design of various algorithms. In our work, we prove a common generalization of this and other known basis exchange properties by showing that for any subsets $X \subseteq A \setminus B$ and $Y \subseteq B \setminus A$, there exist subsets $U \subseteq A \setminus B$ and $V \subseteq B \setminus A$ such that $X\subseteq U$, $Y\subseteq V$, $A-U+V$ and $B+U-V$ are bases, and $|U|=|V|$ is at most the rank of $X+Y$. As an application, we prove the robustness of the local search algorithm for finding maximum weight matroid bases. For matroids representable over a field of characteristic zero, we further generalize our exchange property to include the very recent Equitability Theorem (SODA 2026), by establishing a far-reaching generalization of the Grassmann-Plücker identity.