Generalizing the Multiple Exchange Property for Matroid Bases

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 found applications not only in matroid theory, but also in the analysis and design of various algorithms. This paper generalizes the multiple exchange property in two directions. First, 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 $X+Y$. Second, we develop a general framework for deriving extensions of the Grassmann-Pl\"ucker identity. For matroids representable over fields of characteristic zero, this framework yields new exchange and reconfiguration properties, the latter requiring the resulting basis pair to be reachable from the original one by a sequence of symmetric exchanges. For this matroid class, we obtain an exchange theorem that simultaneously generalizes our first result and the recent Equitability Theorem (SODA 2026). Within the same representable setting, we also derive a weighted equitability theorem, with an application to matroid-constrained EF1 allocations for two agents with additive valuations.