The Max-Flow Min-Cut Theorem for Countable Networks

TL;DR

This paper extends the Max-Flow Min-Cut theorem to countable networks, establishing the existence of orthogonal flow and cut pairs, with special considerations for infinite trails and locally finite networks.

Contribution

It proves a strong version of the Max-Flow Min-Cut theorem for countable networks, including cases with infinite trails and locally finite networks.

Findings

Existence of orthogonal flow and cut pairs in countable networks.

Mundaneness of flows in networks without infinite trails.

Orthogonal pairs satisfying Kirchhoff's law in locally finite networks.

Abstract

We prove a strong version of the Max-Flow Min-Cut theorem for countable networks, namely that in every such network there exist a flow and a cut that are "orthogonal" to each other, in the sense that the flow saturates the cut and is zero on the reverse cut. If the network does not contain infinite trails then this flow can be chosen to be mundane, i.e. to be a sum of flows along finite paths. We show that in the presence of infinite trails there may be no orthogonal pair of a cut and a mundane flow. We finally show that for locally finite networks there is an orthogonal pair of a cut and a flow that satisfies Kirchhoff's first law also for ends.