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10. Network Flows

The chapter delves into the concept of network flows, specifically in the context of linear programming and the Ford-Fulkerson algorithm. It explains the representation of network flows using directed graphs containing source and sink vertices, and highlights the significance of flow conservation and optimization. Additionally, it discusses the relationship between maximum flow and minimum cut, demonstrating how these principles are crucial for efficiently managing resources in a network.

Sections

Network Flows

This section introduces the concept of network flows and their representation in linear programming, focusing on the Ford-Fulkerson algorithm for optimizing flow from a source to a sink in a network.

10 Section Overview

Start current section content and materials

10.1 Bandwidth Allocation Problem

The bandwidth allocation problem involves determining the maximum flow through a network from a source to a sink while adhering to edge capacities.

10.2 Flow Properties

This section discusses the flow properties in network flow problems, focusing on the algorithms used to optimize flow from source to sink in directed graphs.

10.3 Special Graph Type

This section delves into network flow problems, specifically the mechanisms of the Ford-Fulkerson algorithm, its formulation through linear programming, and the importance of maximum flow and minimum cut theorems.

10.4 Setting Up Linear Program

This section explains how to set up a linear program to model network flow problems, particularly focusing on the bandwidth allocation problem.

10.5 Ford-Fulkerson Algorithm

The Ford-Fulkerson algorithm is a method for computing the maximum flow in a flow network.

10.6 Residual Graph

This section introduces the concept of residual graphs used in network flow problems, particularly in the context of the Ford-Fulkerson algorithm.

10.7 Max Flow Min Cut Theorem

The Max Flow Min Cut Theorem states that the maximum flow in a network equals the minimum capacity that can separate the source from the sink.

10.8 Choosing Paths in Ford-Fulkerson

This section discusses the Ford-Fulkerson algorithm for maximizing flow in a network, illustrating how paths are chosen and flows are augmented within a directed graph.

Learning Objectives

  • Network flows are represented in directed graphs with specific source and sink vertices.

  • Flow conservation must ensure that the inflow equals outflow at each internal node of the network.

  • The maximum flow from the source to the sink cannot exceed the capacity defined by the minimum cut in the network.

Key Concepts

Network Flow

A flow in a network is the amount of flow sent from a source node to a sink node through a network of edges, adhering to certain constraints.

Ford-Fulkerson Algorithm

An algorithm used to compute the maximum flow in a flow network by incrementally augmenting paths until no further improvement is possible.

Residual Graph

A transformed version of the original flow graph that reflects the remaining capacities after some flow has been assigned, including backward edges to allow flow adjustment.

Max Flow Min Cut Theorem

A theorem stating that the maximum flow in a network is equal to the capacity of the smallest cut that separates the source and the sink.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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