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Maximum Flow Maximum Flow

Neil TangNeil Tang3/30/20103/30/2010

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Class OverviewClass Overview

The maximum flow problem Applications A greedy algorithm which does not work The Ford-Fulkerson algorithm Implementation and time complexity Another approach: linear programming An Application: maximum matching in a bipartite graph

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The Maximum Flow ProblemThe Maximum Flow Problem

The weight of a link (a.k.a link capacity) indicates the maximum amount of flow allowed to pass through this link.

The maximum flow problem: Given a weighted directed graph G, a source node s and a sink node t, find the maximum amount of flow that can pass from s to t and a corresponding feasible link flow allocation.

Flow feasibility: Both the flow conservation constraint and the capacity constraint must be satisfied.

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The Maximum Flow ProblemThe Maximum Flow Problem

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ApplicationsApplications

Computer networks: Data traffic routing for throughput maximization.

Transportation networks: Road construction and traffic management.

Graph theory: Matching, assignment problems.

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Flow Graph and Residual GraphFlow Graph and Residual Graph

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Basic IdeaBasic Idea

Keep finding s-t augmenting paths until no such paths can be found in the residual graph.

Update the flow and residual graph according to the augmenting path in each step.

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A Greedy Algorithm which Does Not WorkA Greedy Algorithm which Does Not WorkFind an augmenting path s-a-d-t with flow value 3 and update the flow and residual graphs as follows:

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The Ford-Fulkerson AlgorithmThe Ford-Fulkerson AlgorithmFind an augmenting path s-a-d-t with flow value 3 and update the flow and residual graphs as follows:

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The Ford-Fulkerson AlgorithmThe Ford-Fulkerson AlgorithmFind an augmenting path s-b-d-a-c-t with flow value 2 and update the flow and residual graphs as follows:

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The Implementation and Time ComplexityThe Implementation and Time Complexity

If all the link capacities are integers, then the time complexity of the Ford-Fulkerson algorithm is bounded by O(f|E|), where f is the max flow.

A bad example for random path selection.

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The Implementation and Time ComplexityThe Implementation and Time Complexity

In each step, find an augmenting path which allows largest the increase in flow using a modified Dijkstra’s algorithm. It has been proved that it terminates after O(|E|logCapmax) steps, so the time complexity is O(|E|2log|V|logCapmax).

The Edmonds-Karp algorithm: In each step, find an augmenting path with minimum number of edges using BFS. It has been proved that it terminates after O(|E||V|) steps. So the time complexity is O(|E|2|V|).

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Another Approach: Linear Programming Another Approach: Linear Programming

LP in the standard form

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An LP can be solved by existing algorithms in polynomial time.

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Maximum Flow Problem - LPMaximum Flow Problem - LP

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Shortest Path Problem - ILPShortest Path Problem - ILP

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Maximum Matching in A Bipartite GraphMaximum Matching in A Bipartite Graph

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A matching (a.k.a. independent edge set): a set of edges without common

vertices.

The maximum matching problem: find the matching with the maximum number of edges.

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Maximum Matching in A Bipartite GraphMaximum Matching in A Bipartite Graph

CS223 Advanced Data Structures and Algorithms

A max-flow based algorithm