Generalized max-flows and min-cuts in simplicial complexes

TL;DR

This paper extends max-flow and min-cut problems to high-dimensional simplicial complexes, providing algorithms, hardness results, and topological insights, with applications to embedded complexes and algorithmic heuristics.

Contribution

It introduces a topological framework for high-dimensional max-flow/min-cut problems, establishes their computational hardness, and offers algorithms and conditions for specific cases.

Findings

Computing maximum integral flows in complexes is NP-hard.

A natural combinatorial definition of cuts is NP-hard to compute.

Algorithms for complexes embedded in Euclidean space relate to shortest paths and circulations.

Abstract

We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For $d$-dimensional simplicial complexes embedded into $\mathbb{R}^{d+1}$ we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt.