Fair Integral Submodular Flows

TL;DR

This paper studies decreasingly minimal integer-valued elements in integral submodular flow polyhedra, characterizes their structure, and develops a strongly polynomial algorithm for finding such flows, including cost optimization.

Contribution

It introduces a novel characterization of dec-min integral submodular flows as intersections of faces and boxes, enabling efficient computation.

Findings

Dec-min integral elements form a face-intersection of the original polyhedron.

A strongly polynomial algorithm is developed for dec-min and cheapest flows.

Application to strongly connected orientations of mixed graphs.

Abstract

Integer-valued elements of an integral submodular flow polyhedron $Q$ are investigated which are decreasingly minimal (dec-min) in the sense that their largest component is as small as possible, within this, the second largest component is as small as possible, and so on. As a main result, we prove that the set of dec-min integral elements of $Q$ is the set of integral elements of another integral submodular flow polyhedron arising from $Q$ by intersecting a face of $Q$ with a box. Based on this description, we develop a strongly polynomial algorithm for computing not only a dec-min integer-valued submodular flow but even a cheapest one with respect to a linear cost-function. A special case is the problem of finding a strongly connected (or $k$-edge-connected) orientation of a mixed graph whose in-degree vector is decreasingly minimal.