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Ford-Fulkerson terminates for finite integer capacities and proves max-flow min-cut with an integral maximum flow
Statement
In a finite integral - network, begin with the zero flow and repeatedly augment along any residual - path by its integral bottleneck. This process terminates. Its final integral flow is maximum, and
Facts & Assumptions
Given: A finite integral - network and the augmenting-path process from its zero flow.
Residual bottleneck augmentation preserves feasibility and raises the value by its positive integral bottleneck (Augmenting by the residual bottleneck preserves feasibility and increases the flow value).
A flow with no residual - path has a cut of capacity equal to its value (If the residual network has no source-sink path, its reachable set gives a cut whose capacity equals the flow value).
Every feasible flow has value at most every cut capacity (A flow's value is its net flow across every source-sink cut and never exceeds the cut capacity).
Proof
The zero flow is feasible and integral, and every feasible flow has value at most the finite integer sum of capacities of arcs leaving .
Each augmentation increases the integral value by at least one by [L1], so no more than that finite source-capacity bound many augmentations occur.
At termination the final flow has no residual path, so [L2] supplies a cut with .
By [L3], every feasible flow has value at most , and every cut has capacity at least .
Steps 1.1--1.4 prove termination, integrality, maximality of , and equality with the minimum cut capacity.
Remarks
- Integer bottlenecks are the termination mechanism. This theorem makes no assertion about arbitrary real capacities.
Depends on
- Augmenting by the residual bottleneck preserves feasibility and increases the flow value
- If the residual network has no source-sink path, its reachable set gives a cut whose capacity equals the flow value
- A flow's value is its net flow across every source-sink cut and never exceeds the cut capacity
- The cardinality $\lvert A\rvert$ of a finite set
Used by
- A four-vertex integral network with an exhibited maximum flow and minimum cut of value five Example
- The unit network for a bipartite graph identifies integral flows with matchings and finite cuts with vertex covers Proposition
- Menger's theorem: the finite directed and undirected arc, edge and nonadjacent-vertex forms Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 37 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- M. Goemans, Lecture notes on flows and cuts (standard reference, not scraped)