The is a powerful tool in graph theory, linking network flow to . It states that the from to equals the of any cut separating them, revealing bottlenecks and optimizing resource allocation.
This theorem has wide-ranging applications, from transportation and communication networks to and . By modeling scenarios as flow networks, we can solve complex problems in scheduling, resource allocation, and even .
Understanding the Min-Cut Max-Flow Theorem
Min-cut max-flow theorem fundamentals
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Min-Cut Max-Flow Theorem states maximum flow from source to sink equals minimum capacity of any cut separating source from sink in any network
Duality between flow and cut problems reveals upper bound on maximum flow and helps identify bottlenecks in network flow
Flow represents amount moving through network (water in pipes)
Cut partitions vertices into two disjoint subsets (North and South hemispheres)
Capacity limits maximum amount flowing through an edge (pipe diameter)
Theorem applies to (highway systems), (internet traffic), and resource allocation (supply chains)
Proof of min-cut max-flow theorem
Proof begins with maximum flow in network and constructs minimum cut using residual graph
Define set S containing source and all vertices reachable in residual graph
S and its complement form a cut with no augmenting paths in residual graph
Flow across cut equals its capacity due to and capacity constraints
Constructed cut has capacity equal to maximum flow, no cut can have smaller capacity