A Simpler and Faster Strongly Polynomial Algorithm for Generalized Flow Maximization

TL;DR

This paper introduces a new strongly polynomial algorithm for generalized flow maximization that is simpler, faster, and primarily works with integral flows, significantly improving efficiency over previous methods.

Contribution

The paper presents a novel strongly polynomial algorithm that relaxes primal feasibility, enabling the use of integral flows and achieving substantial complexity improvements.

Findings

Complexity bound improved by almost a factor of O(n^2)

Algorithm works efficiently even for small numerical parameters

Primarily uses integral flows, simplifying previous approaches

Abstract

We present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm [V\'egh16]. For the uncapacitated problem formulation, the complexity bound $O(mn(m+n\log n)\log (n^2/m))$ improves on the previous estimate by almost a factor $O(n^2)$. Even for small numerical parameter values, our running time bound is comparable to the best weakly polynomial algorithms. The key new technical idea is relaxing the primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms for the problem.