The Maxflow problem and a generalization to simplicial complexes

TL;DR

This paper explores a generalization of the Maxflow problem within the framework of simplicial complexes, aiming to extend existing algorithms and understanding from graph theory to higher-dimensional structures.

Contribution

It introduces a novel generalization of the Maxflow problem to simplicial complexes, connecting classical graph algorithms to higher-dimensional topological structures.

Findings

Reduction of the simplicial complex Maxflow problem to the classical graph case

Potential for extending efficient Maxflow algorithms to higher dimensions

Framework for analyzing flow problems in complex topological spaces

Abstract

The problem of Maxflow is a widely developed subject in modern mathematics. Efficient algorithms exist to solve this problem, that is why a good generalization may permit these algorithms to be understood as a particular instance of solutions in a wider class of problems. In the last section we suggest a generalization in the context of simplicial complexes, that reduces to the problem of Maxflow in graphs, when we consider a graph as a simplicial complex of dimension 1.