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A Mechanized Proof of the Max-Flow Min-Cut Theorem for Countable Networks with Applications to Probability Theory

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Abstract

Aharoni et al. (J Combinat Theory, Ser B 101:1–17, 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 and the sink is finite. This proof is based on the max-flow min-cut theorem for finite networks. As a use case, we formalize a characterization theorem for relation lifting on discrete probability distributions and two of its applications.

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Notes

  1. The record package achieves extensibility with structural subtyping by internally generalizing \(\alpha \ \textsf {graph}\) to \((\alpha , \beta )\ \textsf {graph-scheme}\), where \(\beta \) is the extension slot for further fields. For example, \(\beta \) is instantiated with the singleton type \(\textsf {unit}\) for \(\textsf {graph}\). All operations on \(\textsf {graph}\) are actually defined on \(\textsf {graph-scheme}\) so that they also work for all record extensions. We omit this technicality from the presentation.

  2. Sack and Zhang set \(e_{i,j} = \infty \) if \(x_i \mathrel {R} y_j\), but the max-flow min-cut theorem handles only finite edge capacities. Their argument works unchanged for any value greater than 1, such as our choice of 2.

  3. Sack and Zhang’s proof is formalized in the theory Rel_PMF_Characterisation in the accompanying AFP entry [25] version for Isabelle2016-1. The modified proof can be found in the theory Rel_PMF_Characterisation_MFMC in the version for Isabelle2021-1 of the same AFP entry.

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Acknowledgements

Swiss National Science Foundation Grant 153217 “Formalising Computational Soundness for Protocol Implementations.” This work was partially done while the author was at ETH Zurich. 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.

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    Andreas Lochbihler

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Lochbihler, A. A Mechanized Proof of the Max-Flow Min-Cut Theorem for Countable Networks with Applications to Probability Theory. J Autom Reasoning 66, 585–610 (2022). https://doi.org/10.1007/s10817-022-09616-4

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