[Verse 1]
Picture networks where the water rushes through
Source to sink, channels narrow, channels new
Every edge can carry limited amounts
Bottlenecks decide what truly counts
Ford and Fulkerson mapped the territory
Maximum throughput meets minimum boundary
Push and pull until saturation's reached
Sacred balance that the masters preached
[Chorus]
Max equals min, the theorem stands supreme
Capacity cuts block the flowing stream
When augmenting paths have disappeared
Maximum flow and minimum cut are mirrored
Max equals min, duality revealed
What flows through equals what gets sealed
[Verse 2]
Residual graphs expose the hidden space
Backward edges show reverse's grace
Saturated edges vanish from the scene
While unused capacity keeps the pipeline clean
Cut partitions split the vertex set in two
Source on left and sink on right side's view
Sum those forward edges crossing the divide
That's your bottleneck that can't be denied
[Chorus]
Max equals min, the theorem stands supreme
Capacity cuts block the flowing stream
When augmenting paths have disappeared
Maximum flow and minimum cut are mirrored
Max equals min, duality revealed
What flows through equals what gets sealed
[Bridge]
Edmonds-Karp refines the algorithm's speed
Breadth-first searching plants the greedy seed
Polynomial time complexity achieved
Network optimization finally relieved
From exponential nightmares to efficient code
Linear programming's dual bestowed
[Verse 3]
Applications span from matching to supply chains
Bipartite graphs where perfect pairing reigns
Image segmentation cuts through pixel noise
Network reliability makes the optimal choice
Transportation problems find their cheapest routes
While minimum vertex cuts compute disputes
Theory proves what intuition suspected all along
Maximum throughput sings the minimum's song
[Chorus]
Max equals min, the theorem stands supreme
Capacity cuts block the flowing stream
When augmenting paths have disappeared
Maximum flow and minimum cut are mirrored
Max equals min, duality revealed
What flows through equals what gets sealed
[Outro]
When the network reaches equilibrium state
No more augmentation can alter fate
The tightest constraint determines the bound
Maximum flow and minimum cut, forever crowned