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Network flows are a powerful tool in combinatorial optimization, modeling various real-world problems as directed graphs with capacities. They're used to find optimal solutions in transportation, communication, and resource allocation scenarios.

The maximum flow problem aims to push the most "flow" from source to sink, respecting edge capacities. Its dual, the minimum cut problem, finds the smallest capacity cut separating source and sink. These concepts are crucial for understanding network optimization techniques.

Network flows and components

Definition and key components

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  • A network flow is a directed graph with a source node and a sink node, where each edge has a capacity representing the maximum amount of flow that can pass through it
  • The capacity constraint states that the flow along an edge cannot exceed its capacity
  • The conservation constraint requires that, for each node except the source and sink, the total flow entering the node must equal the total flow leaving the node
  • A feasible flow satisfies both the capacity and conservation constraints
  • The value of a flow is the total amount of flow passing from the source to the sink

Examples of network flows

  • Transportation networks (road systems, rail networks) can be modeled as flow networks to optimize traffic flow and minimize congestion
  • In communication networks, network flows can be used to determine the maximum data transmission rate between two nodes and identify bottlenecks
  • Resource allocation problems (assigning tasks to workers, distributing goods from warehouses to retailers) can be solved using network flow techniques
  • Bipartite matching, a special case of network flow, can be used to solve problems like assigning students to projects or matching organ donors to recipients

Maximum flow vs minimum cut

Maximum flow problem

  • The maximum flow problem aims to find the largest possible flow from the source to the sink in a network, while respecting the capacity and conservation constraints
  • The value of the maximum flow represents the maximum amount of flow that can be sent through the network from the source to the sink
  • Finding the maximum flow helps identify the bottlenecks and the overall capacity of the network

Minimum cut problem

  • A cut in a flow network is a partition of the nodes into two disjoint subsets, with the source in one subset and the sink in the other
  • The capacity of a cut is the sum of the capacities of the edges from the source's subset to the sink's subset
  • The minimum cut problem aims to find the cut with the smallest capacity among all possible cuts in the network
  • The max-flow min-cut theorem states that the maximum flow in a network is equal to the minimum capacity of all cuts in the network
  • The minimum cut problem is the dual of the maximum flow problem, as solving one problem automatically solves the other

Algorithms for maximum flow

Ford-Fulkerson algorithm

  • The Ford-Fulkerson algorithm is a greedy approach to finding the maximum flow in a network by iteratively augmenting the flow along augmenting paths
  • An augmenting path is a path from the source to the sink in the residual graph, which is constructed by subtracting the current flow from the original capacities and adding reverse edges with the current flow as their capacity
  • The algorithm starts with an initial flow of zero and iteratively finds an augmenting path in the residual graph
  • The flow is then increased along the augmenting path by the minimum residual capacity of the edges in the path
  • The algorithm terminates when no more augmenting paths can be found, and the final flow is the maximum flow

Edmonds-Karp algorithm

  • The Edmonds-Karp algorithm is an implementation of the Ford-Fulkerson algorithm that uses breadth-first search to find the shortest augmenting path in each iteration
  • By always choosing the shortest augmenting path, the Edmonds-Karp algorithm guarantees that the number of iterations is bounded by O(VE)O(VE), where VV is the number of vertices and EE is the number of edges in the network
  • The time complexity of the Edmonds-Karp algorithm is O(VE2)O(VE^2), as each iteration takes O(E)O(E) time to find the shortest augmenting path using breadth-first search
  • The Edmonds-Karp algorithm is more efficient than the general Ford-Fulkerson algorithm, which may take exponential time if the augmenting paths are chosen poorly

Applications of network flows

Transportation and communication networks

  • Transportation networks (road systems, rail networks) can be modeled as flow networks to optimize traffic flow and minimize congestion
  • Maximum flow algorithms can be used to identify bottlenecks and determine the maximum capacity of the transportation network
  • In communication networks, maximum flow algorithms can be used to determine the maximum data transmission rate between two nodes and identify bottlenecks
  • Network flow techniques can help optimize the design and operation of communication networks to ensure efficient data transmission

Resource allocation and matching problems

  • Resource allocation problems (assigning tasks to workers, distributing goods from warehouses to retailers) can be solved using network flow techniques
  • The maximum flow algorithm can be used to find the optimal assignment of resources to maximize the overall efficiency or minimize the total cost
  • Bipartite matching, a special case of network flow, can be used to solve problems like assigning students to projects or matching organ donors to recipients
  • The maximum flow algorithm can be applied to find the maximum number of matchings in a bipartite graph, ensuring the best possible allocation of resources

Image segmentation

  • Network flow algorithms can be applied to solve problems in image segmentation, where the goal is to partition an image into meaningful regions based on pixel similarities
  • The image is represented as a graph, with pixels as nodes and edges connecting similar pixels
  • A source node is connected to the foreground pixels, and a sink node is connected to the background pixels
  • The maximum flow algorithm is then used to find the minimum cut, which separates the foreground from the background, resulting in the segmented image
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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