Network Flow and Maximum-Flow Minimum-Cut Theorem Interactive Widget

The tightest squeeze sets the maximum

Every network has a bottleneck — the narrowest point that limits total flow. Surprisingly, the max flow you can push through equals the capacity of the tightest cut you can make. Drag the slider to watch flow pile up at the network's true limit.

Network Flow
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Bottleneck at cut: 0 / 0 capacity
Attempted Flow
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Actual Flow
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Min Cut Capacity
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The max-flow min-cut theorem states that the maximum amount of flow you can push from source to sink equals the minimum capacity of any cut separating them. A cut is a partition of the network into two sets — one containing the source, one containing the sink — and its capacity is the sum of edge capacities crossing from source side to sink side. Why does this work? Because flow is conserved at every node (what comes in must go out), so all flow must cross any cut you make. The tightest cut is therefore the absolute ceiling. This principle applies to water pipes, road networks, data transmission, electrical grids, and evacuation routes — the bottleneck always wins.
Know This
The maximum flow through any network always equals the capacity of its minimum cut, because every unit of flow must cross every cut separating source from sink.