Decreasing Minimization on M-convex Sets: Background and Structures

TL;DR

This paper explores the structure and properties of decreasingly-minimal elements in M-convex sets, revealing their rich structure, connections to matroids, and applications in resource allocation and graph orientation.

Contribution

It characterizes the set of dec-min elements in M-convex sets, showing their structure via a canonical chain and linking them to matroids, with implications for algorithms and applications.

Findings

Dec-min elements form a structured set described by a canonical chain.

Dec-min elements are characterized by minimizing the sum of squares of components.

The set of dec-min elements arises from matroids via characteristic vector translation.

Abstract

The present work is the first member of a pair of papers concerning decreasingly-minimal (dec-min) elements of a set of integral vectors, where a vector is dec-min if its largest component is as small as possible, within this, the next largest component is as small as possible, and so on. This discrete notion, along with its fractional counterpart, showed up earlier in the literature under various names. The domain we consider is an M-convex set, that is, the set of integral elements of an integral base-polyhedron. A fundamental difference between the fractional and the discrete case is that a base-polyhedron has always a unique dec-min element, while the set of dec-min elements of an M-convex set admits a rich structure, described here with the help of a "canonical chain". As a consequence, we prove that this set arises from a matroid by translating the characteristic vectors of its bases with an integral vector. By relying on these characterizations, we prove that an element is dec-min if and only if the square-sum of its components is minimum, a property resulting in a new type of min-max theorems. The characterizations also give rise, as shown in the companion paper, to a strongly polynomial algorithm, and to several applications in the areas of resource allocation, network flow, matroid, and graph orientation problems, which actually provided a major motivation to the present investigations. In particular, we prove a conjecture on graph orientation.