# Network Simplex Algorithm associated with the Maximum Flow Problem

## TL;DR

This paper adapts the network simplex algorithm for maximum flow problems, demonstrating that it avoids cycling issues that cause infinite loops in the minimum cost flow context.

## Contribution

It introduces a version of the network simplex algorithm tailored for maximum flow problems and proves the absence of cycling phenomena.

## Key findings

- The algorithm successfully solves maximum flow problems.
- Cycling phenomena do not occur in this adapted algorithm.
- The method improves understanding of algorithmic behavior in flow problems.

## Abstract

In the present paper, we apply the network simplex algorithm for solving the minimum cost flow problem, to the maximum flow problem. Then we prove that the cycling phenomenon which causes the infinite loop in the algorithm, does not occur in the network simplex algorithm associated with the maximum flow problem.