9.1 Maximum flow problem and Ford-Fulkerson algorithm
2 min read•july 24, 2024
Network theory is all about maximizing flow through a system. The finds the most stuff we can push from a start point to an end point, following rules about how much each connection can handle.
The solves this by repeatedly finding paths to send more flow until we can't anymore. It's used in real-world situations like water systems and power grids to figure out the most efficient way to move resources around.
Understanding the Maximum Flow Problem
Components of maximum flow problem
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Maximum flow problem optimizes network flow theory finds maximum amount of flow through network
Network components include (s) starts flow, (t) ends flow, edges connect nodes, nodes serve as vertices
c(u,v) limits maximum flow on edge (u,v)
Flow f(u,v) represents amount passing through edge (u,v)
ensure capacity constraint 0≤f(u,v)≤c(u,v) and flow conservation (incoming equals outgoing except source and sink)
shows remaining available capacity on edges (transportation networks, communication systems)
Ford-Fulkerson Algorithm and Its Application
Steps of Ford-Fulkerson algorithm
Initialize flow to zero for all edges
Find from source to sink in residual network
Determine bottleneck capacity along augmenting path
Update flow along augmenting path
Update residual network
Repeat steps 2-5 until no augmenting path exists
Augmenting path connects source to sink in residual network with available capacity
Bottleneck capacity represents minimum residual capacity along augmenting path
Algorithm terminates when no more augmenting paths exist in residual network (water supply systems, electrical grids)
Application of Ford-Fulkerson algorithm
Draw initial network with capacities
Create residual network
Find augmenting path using depth-first search or breadth-first search
Identify bottleneck capacity
Update flow and residual network
Repeat until no augmenting path exists
Applies to simple networks with few nodes and edges, complex networks with multiple intermediate nodes, networks with parallel edges, networks with bidirectional flows
Special cases include networks with multiple sources or sinks, networks with edge lower bounds (transportation routing, supply chain optimization)
Time complexity of Ford-Fulkerson algorithm
Worst-case time complexity O(E⋅∣fmax∣), E represents number of edges, ∣fmax∣ denotes maximum flow value
Time complexity depends on number of augmenting paths and choice of augmenting path selection method
Edmonds-Karp algorithm uses breadth-first search, time complexity O(VE2), V represents number of vertices
improves with blocking flows, time complexity O(V2E)
Practical considerations include performance on sparse vs dense networks and impact of network structure on iterations (computer networks, social network analysis)