An Algorithmic Upper Bound for Permanents via a Permanental Schur Inequality

Abstract: Computing the permanent of a non-negative matrix is a computationally challenging, #P-complete problem with wide-ranging applications. We introduce a novel permanental analogue of Schur's determinant formula, leveraging a newly defined \emph{permanental inverse}. Building on this, we introduce an iterative, deterministic procedure called the \emph{permanent process}, analogous to Gaussian elimination, which yields constructive and algorithmically computable upper bounds on the permanent. Our framework provides particularly strong guarantees for matrices exhibiting approximate diagonal dominance-like properties, thereby offering new theoretical and computational tools for analyzing and bounding permanents.