On stable flows and preflows

TL;DR

This paper introduces a new polynomial-time algorithm for finding stable flows in networks with multiple sources and sinks, using preflows, and extends the concept to stable quasi-flows with bounded excesses.

Contribution

It presents an alternative $O(nm)$ algorithm for stable flows based on preflows and generalizes to stable quasi-flows with bounded excesses.

Findings

The algorithm runs in $O(nm)$ time.

It generalizes to stable quasi-flows with bounded excesses.

Provides an efficient method for stable flow computation.

Abstract

In 2010s Fleiner introduced a notion of stable flows in directed networks and showed that such a flow always exists and can be found by use of a reduction to the stable allocation problem due to Baiou and Balinski. Recently Cseh and Matuschke devised a direct strongly polynomial algorithm. In this paper we give an alternative algorithm to find a stable flow in a network with several sources and sinks. It is based on an idea of preflows (appeared in 1970s in a faster algorithm for the classical max-flow problem), and runs in $O(nm)$ time for a network with $n$ vertices and $m$ edges. The results are further generalized to a larger class of objects, so-called stable quasi-flows with bounded excesses in non-terminal vertices. (The paper is written in Russian.)