# Fair Integral Network Flows

## TL;DR

This paper introduces a strongly polynomial algorithm for finding a fair, decreasingly minimal integer-valued $st$-flow in a network, optimizing flow distribution on a subset of edges to ensure fairness.

## Contribution

It develops a novel strongly polynomial algorithm for computing decreasingly minimal flows, providing a formal and computational approach to fairness in network flows.

## Key findings

- Algorithm guarantees a strongly polynomial runtime.
- Successfully computes fair flow distributions on specified edges.
- Provides a characterization of decreasingly minimal flows.

## Abstract

A strongly polynomial algorithm is developed for finding an integer-valued feasible $st$-flow of given flow-amount which is decreasingly minimal on a specified subset $F$ of edges in the sense that the largest flow-value on $F$ is as small as possible, within this, the second largest flow-value on $F$ is as small as possible, within this, the third largest flow-value on $F$ is as small as possible, and so on. A characterization of the set of these $st$-flows gives rise to an algorithm to compute a cheapest $F$-decreasingly minimal integer-valued feasible $st$-flow of given flow-amount. Decreasing minimality is a possible formal way to capture the intuitive notion of fairness.