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# A Mechanized Proof of the Max-Flow Min-Cut Theorem for Countable Networks

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## Acknowledgements

We thank Ron Aharoni and Eli Berger for helping to clarify the weaknesses in the original proofs. S. Reza Sefidgar and the anonymous reviewers helped to improve the presentation.

## Cite As

Andreas Lochbihler. A Mechanized Proof of the Max-Flow Min-Cut Theorem for Countable Networks. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITP.2021.25

## Abstract

Aharoni et al. [Ron Aharoni et al., 2010] proved the max-flow min-cut theorem for countable networks, namely that in every countable network with finite edge capacities, there exists a flow and a cut such that the flow saturates all outgoing edges of the cut and is zero on all incoming edges. In this paper, we formalize their proof in Isabelle/HOL and thereby identify and fix several problems with their proof. We also provide a simpler proof for networks where the total outgoing capacity of all vertices other than the source is finite. This proof is based on the max-flow min-cut theorem for finite networks.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Network flows
• Theory of computation → Higher order logic
• Theory of computation → Logic and verification
##### Keywords
• flow network
• optimization
• infinite graph
• Isabelle/HOL

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## References

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