Prefix-bounded matrices

TL;DR

This paper introduces prefix-bounded matrices (PBMs), unifies their properties with earlier matrix classes, and demonstrates their convex hulls form intersections of generalized polymatroids, confirming conjectures and enabling efficient algorithms.

Contribution

It unifies various ASM extensions into prefix-bounded matrices, proves their convex hulls are TDI, and links their linear systems to network matrices for algorithmic applications.

Findings

Convex hull of PBMs is the intersection of two generalized polymatroids.

The linear inequality system for ASMs is totally dual integral (TDI).

The convex hull of PBMs has the integer decomposition property.

Abstract

By unifying various earlier extensions of alternating sign matrices (ASMs), we introduce the notion of prefix-bounded matrices (PBMs). It is shown that the convex hull of these matrices forms the intersection of two special generalized polymatroids. This implies $\unicode{x2013}$ in a more general form $\unicode{x2013}$ that the linear inequality system given by Behrend and Knight (2007) and by Striker (2007, 2009) for describing the polytope of alternating sign matrices is totally dual integral (TDI), confirming a recent conjecture of Edmonds (2024, 2025). By relying on the polymatroidal approach, we derive a characterization for the existence of prefix-bounded matrices meeting lower and upper bounds on their entries. Furthermore, we point out that the constraint matrix of the linear system describing the convex hull of PBMs, in particular ASMs, is a network matrix. This implies that (a) standard network-flow techniques can be used to manage algorithmically optimization and structural results on PBMs obtained via g-polymatroids, (b) the linear system is actually box-TDI, and (c) the convex hull of PBMs admits a sharpened form of the integer Carath\'eodory property, in particular, the integer decomposition property. This latter feature makes it possible to confirm an extended form of an elegant conjecture of Brualdi and Dahl (2023) on the decomposability of a so-called $k$-regular alternating sign matrix as the sum of $k$ pattern-disjoint ASMs.