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### Abstract

This article formalises a proof of the maximum-flow minimal-cut
theorem for networks with countably many edges. A network is a
directed graph with non-negative real-valued edge labels and two
dedicated vertices, the source and the sink. A flow in a network
assigns non-negative real numbers to the edges such that for all
vertices except for the source and the sink, the sum of values on
incoming edges equals the sum of values on outgoing edges. A cut is a
subset of the vertices which contains the source, but not the sink.
Our theorem states that in every network, there is a flow and a cut
such that the flow saturates all the edges going out of the cut and is
zero on all the incoming edges. The proof is based on the paper
The Max-Flow Min-Cut theorem for countable networks by
Aharoni et al. Additionally, we prove a characterisation of the
lifting operation for relations on discrete probability distributions,
which leads to a concise proof of its distributivity over relation
composition.

### License

### History

- August 13, 2021
- generalize the derivation of the characterisation for the relator of discrete probability distributions to work for the bounded and unbounded MFMC theorem
(revision 3c85bb52bbe6)
- December 19, 2020
- simpler proof of linkability for bounded unhindered bipartite webs, leading to a simpler proof for networks with bounded out-capacities
(revision 93ca33f4d915)

- September 6, 2017
- derive characterisation for the lifting operation on discrete distributions from finite version of the max-flow min-cut theorem
(revision a7a198f5bab0)

### Topics

### Session MFMC_Countable

- MFMC_Misc
- MFMC_Finite
- Matrix_For_Marginals
- MFMC_Network
- MFMC_Web
- MFMC_Reduction
- MFMC_Bounded
- MFMC_Flow_Attainability
- MFMC_Unbounded
- Max_Flow_Min_Cut_Countable
- Rel_PMF_Characterisation
- Rel_PMF_Characterisation_MFMC