7.2 Maximum flow and minimum cut problems

Network flow problems model resource distribution through systems with capacity constraints. The maximum flow problem aims to optimize the total flow from a source to a sink node, respecting edge capacities and flow conservation at intermediate nodes.

The Ford-Fulkerson algorithm solves max-flow problems by iteratively finding augmenting paths. Cut theory complements this approach, showing that the maximum flow equals the capacity of the minimum cut separating source and sink, with applications in network analysis and optimization.

Network Flow Fundamentals

Formulation of maximum flow problems

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• Network structure represented as directed graph with source and sink nodes, edges with capacities (water pipeline system)
• Flow conservation ensures inflow equals outflow at all nodes except source and sink maintains balance
• Capacity constraints limit flow on each edge to its capacity prevents overflow
• Non-negativity constraints keep flow values positive reflects physical reality
• Objective function maximizes total flow from source to sink optimizes network utilization

Application of Ford-Fulkerson algorithm

• Algorithm steps:

1. Find augmenting path from source to sink
2. Determine residual capacity of path
3. Augment flow along path
4. Update residual graph
5. Repeat until no augmenting path exists

• Residual graph represents remaining capacity on edges includes forward and backward edges
• Augmenting path connects source to sink in residual graph uses forward and backward edges
• Termination condition reached when no augmenting path exists in residual graph
• Time complexity $O(E \cdot f_{max})$ where $E$ is number of edges and $f_{max}$ is maximum flow value

Cut Theory and Applications

Minimum cuts in networks

• Cut partitions nodes into two disjoint sets with source and sink in different sets (communication network)
• Cut capacity sums capacities of edges crossing from source side to sink side
• Minimum cut has smallest capacity among all possible cuts critical for network analysis
• Relationship to maximum flow value equals capacity of minimum cut fundamental theorem
• Saturated edges with flow equal to capacity in maximum flow solution form minimum cut when removed

Max-flow min-cut theorem applications

• Max-flow min-cut theorem states maximum flow value equals minimum cut capacity
• Ford-Fulkerson algorithm finds maximum flow in network
• Minimum cut identified by breadth-first search in residual graph from source
• Cut capacity calculated by summing capacities of edges from reachable to unreachable nodes
• Verification ensures cut capacity equals maximum flow value
• Applications include network reliability analysis, image segmentation (medical imaging), bipartite matching (job assignments)