A Structure Theory for the Parametric Submodular Intersection Problem

Kiyohito NAGANO\dagger (11Lnnnnn!!!llll)))BggX) Abstract A linearly parameterized polymatroid intersection problem appears in the context of principal partitions. We consider a submodular intersection problem on a pair of strong-map sequences of submodular functions, which is an extension of the linearly parameterized polymatroid intersection problem to a nonlinearly parameterized one. We introduce the concept of a basis-frame on a finite nonempty set $V$ that gives a mapping from the set of all base polyhedra in $\mathbb{R}^{V}$ into $\mathbb{R}^{V}$ such that each base polyhe-dron in $\mathbb{R}^{V}$ is mapped to one of its bases. We show the existence of a simple universal representation of all optimal solutions of the parameterized submodular intersection problem by means of basis-frames.