Caged Retractions of Polymatroids

Abstract

We develop a unified theory of caged retractions of discrete polymatroids. Given a polymatroid and a cage $\kappa$, the $\kappa$-retraction is a canonical $\kappa$-caged polymatroid obtained by projecting bases into the cage and retaining the maximal projected bases. We prove that this construction agrees with an explicit rank-function formula. We show that the inclusion of the $\kappa$-caged polymatroids into all polymatroids and the $\kappa$-retraction form a Galois connection with respect to the weak-map order. As applications, we obtain caged versions of polymatroid union, the disjoint basis theorem, and induction along a bipartite graph. When $\kappa=\textbf{1}$, these recover the corresponding matroid constructions. We also study how caged retractions interact with Lorentzian polynomials and representations over near-idempotent tracts. In each case, the construction preserves the relevant structure.