[Verse 1]
Network flows like water through the pipes
Source to sink, capacity defines our types
Ford and Fulkerson cracked the riddle clean
Maximum flow equals minimum cut scene
Build a graph with edges weighted tight
Push the liquid till you reach the height
Residual networks show what's left behind
Augmenting paths reveal what you can find
[Chorus]
Max equals min, that's the theorem's call
Cut the bottleneck, watch the liquid fall
Flow conservation at every single node
Max-flow min-cut, algorithms decode
What goes in must equal what comes out
Maximum throughput, that's what it's about
[Verse 2]
Start with zero flow, build it step by step
Find an augmenting path, keep your network prepped
Residual capacity shows remaining space
Backward edges help you change your pace
When no more paths exist from start to end
Current flow is max, that's when algorithms bend
The cut that blocks this flow has minimum weight
Duality proven, sealed by theorem's fate
[Chorus]
Max equals min, that's the theorem's call
Cut the bottleneck, watch the liquid fall
Flow conservation at every single node
Max-flow min-cut, algorithms decode
What goes in must equal what comes out
Maximum throughput, that's what it's about
[Bridge]
Edmonds-Karp improves the basic scheme
Breadth-first search maintains the flow regime
Polynomial time complexity achieved
Network problems finally relieved
Bipartite matching, airline scheduling too
Image segmentation, database queries through
Applications endless when you understand
Max-flow min-cut rules the algorithm land
[Chorus]
Max equals min, that's the theorem's call
Cut the bottleneck, watch the liquid fall
Flow conservation at every single node
Max-flow min-cut, algorithms decode
What goes in must equal what comes out
Maximum throughput, that's what it's about
[Outro]
Capacity constraints and conservation laws
Network optimization without any flaws
When the maximum flow meets the minimum cut
Ford-Fulkerson theorem keeps the pathway shut