# The Pseudoflow Algorithm and the Pseudoflow-Based Simplex for the Maximum Flow Problem

## Abstract

We introduce an algorithm that solves the maximum flow problem without generating flows explicitly. The algorithm solves di- rectly a problem we call the maximum *s*-excess problem. That problem is equivalent to the minimum cut problem, and is a direct extension of the maximum closure problem. The concepts used also lead to a new parametric analysis algorithm generating all breakpoints in the amount of time of a single run.

The insights derived from the analysis of the new algorithm lead to a new simplex algorithm for the maximum flow problem — a pseudoflow-based simplex. We show that this simplex algorithm can perform a parametric analysis in the same amount of time as a single run. This is the first known simplex algorithm for maximum flow that generates all possible breakpoints of parameter values in the same complexity as required to solve a single maximum flow instance and the fastest one.

The complexities of our pseudoflow algorithm, the new simplex algo- rithm, and the parametric analysis for both algorithms are *O*(*mn*log *n*) on a graph with *n* nodes and *m* arcs.

## Keywords

Sink Node Extended Network Simplex Algorithm Balance Constraint Maximum Flow Problem## Preview

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