Maximum Flow Algorithms — Ford-Fulkerson, Edmonds-Karp & Bipartite Matching | Chapter 24 in Introduction to Algorithms

Chapter 24 of Introduction to Algorithms (CLRS) tackles the maximum-flow problem—how to move the greatest possible amount of material through a network from a source to a sink without exceeding edge capacities. This problem has critical applications in transportation, communication networks, and resource allocation. The chapter introduces essential flow concepts, residual networks, augmenting paths, and foundational algorithms like Ford-Fulkerson and Edmonds-Karp. It also explains how flow models apply to bipartite matching and special network structures.

📺 Watch the full chapter summary above, or continue reading for a complete breakdown of every core concept and algorithm covered in Chapter 24.

What Is a Flow Network?

A flow network is a directed graph where each edge has a nonnegative capacity, and the goal is to determine how much material can be pushed from a source node s to a sink node t while respecting capacity and conservation constraints.

• Capacity constraint: 0 ≤ f(u, v) ≤ c(u, v)
• Flow conservation: For all nodes except s and t, incoming flow = outgoing flow
• Value of a flow: Total flow out of s = Total flow into t

Residual Networks and Augmenting Paths

To find maximum flow, we create a residual network that shows the remaining capacity along each edge and allows us to adjust flow using augmenting paths.

• Residual capacity: c_f(u, v) = remaining capacity on edge (u, v)
• Residual network G_f: Includes forward edges with capacity and reverse edges for possible flow reduction
• Augmenting path: A path from s to t in G_f with positive residual capacities
• Flow update: Increase flow by the minimum residual capacity on the path

Cuts and the Max-Flow Min-Cut Theorem

A cut partitions the graph into two sets (S, T), where s ∈ S and t ∈ T. This leads to one of the most profound theorems in algorithmic graph theory:

• Cut capacity: Sum of capacities of edges from S to T
• Max-Flow Min-Cut Theorem: A flow is maximum if and only if there is no augmenting path. The value of the maximum flow equals the capacity of the minimum cut.

Ford-Fulkerson Method

Ford-Fulkerson is a greedy algorithm that repeatedly finds augmenting paths and adjusts the flow. It is simple but powerful:

• Works best when all capacities are integers
• Each iteration increases the flow by at least 1 unit
• Time complexity: O(E × |f*|), where |f*| is the value of the max flow
• May not terminate if irrational capacities exist

Edmonds-Karp Algorithm

Edmonds-Karp is a more refined version of Ford-Fulkerson that uses BFS to always choose the shortest augmenting path (in number of edges):

• Ensures faster convergence and polynomial time
• Distance from s to each node increases monotonically
• Time complexity: O(VE²)
• Applications: Especially effective in bipartite matching and bounded flow systems

Handling Special Network Structures

CLRS explains how to adapt maximum flow algorithms for more complex scenarios:

• Antiparallel edges: Use a dummy node to separate bidirectional flow
• Multiple sources/sinks:
  • Add a supersource connected to all sources
  • Add a supersink connected from all sinks
  • Use ∞ capacity edges for modeling

Application: Maximum Bipartite Matching

Bipartite matching is a classic example where the maximum flow model shines:

• Given a bipartite graph G = (L ∪ R, E)
• Add source s → connect to all L nodes
• Add sink t ← connect from all R nodes
• Edges from L to R have capacity = 1
• Result: Max flow value = size of the max matching
• Time complexity: O(VE) with Edmonds-Karp

Glossary of Key Terms

• Flow Network: Graph with source, sink, and edge capacities
• Flow (f): Quantity of material sent along each edge
• Residual Capacity: Remaining capacity for sending more flow
• Residual Network (G_f): Graph showing flow potential in current configuration
• Augmenting Path: A valid path from source to sink in G_f
• Max-Flow Min-Cut Theorem: Max flow = Min cut capacity
• Ford-Fulkerson: Simple method for finding max flow
• Edmonds-Karp: BFS-enhanced Ford-Fulkerson with polynomial time guarantee
• Bipartite Matching: Maximum pairing of two disjoint sets using flow logic
• Integrality Theorem: If capacities are integers, so is the resulting max flow

Conclusion

Chapter 24 introduces students to the power and elegance of network flow algorithms. Whether solving matching problems or optimizing traffic and logistics, algorithms like Ford-Fulkerson and Edmonds-Karp form the core of algorithmic reasoning in directed flow networks. With practical applications in real-world systems, understanding flow conservation, augmenting paths, and residual networks is essential for mastering advanced algorithm design.

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