Module 5 – Networks and Decision Mathematics Chapter 24 – Directed Graphs

24.2 Network Flows 0 An application of weighted digraphs involves the concept of flows. 0 A flow is the quantity of material that can move along a given channel; for example, traffic flow along a highway or water flow through a pipe.

Flow Capacities 0 The network’s starting node(s) is called the source. This is where all flows commence. 0 The flow goes through the network to the end node(s) which is called the sink. 0 The flow capacity (capacity) of an edge is the amount of flow that an edge can allow through if it is not connected to any other edges.

0 The inflow of a node is the total of the flows of all edges leading into the node. 0 The outflow of a node is the minimum value obtained when one compares the inflow to the sum of the capacities of all the edges leaving the node.

Consider the following figures:

Example – Maximum Flow by inspection Flights connect the airports at Sydney, Melbourne, Brisbane, Adelaide and Canberra. The numbers in this graph represent the maximum number of passengers that can be carried in a fixed time. Note: There are no direct flights between Melbourne and Brisbane or between Sydney and Adelaide in the time period considered. a) By inspection: Find the maximum number of passengers that can be carried between Sydney and Adelaide (in the fixed time). b) Indicate how this can be achieved.

Solution: a) The maximum number between Sydney and Adelaide is = 1700 b) 700 passengers through Brisbane to Adelaide passengers to Melbourne and of these 600 go directly to Adelaide and an additional 400 go via Canberra. The maximum flow through most simple networks can be determined using inspection, but more complex networks require differing methods to be used….

Minimum cut – Maximum flow method 0 To determine the maximum flow, the network first needs to be divided or ‘cut’ into two parts. A cut in a network diagram is a line drawn through a number of edges which stops all flow from the source to the sink (it is defined as a collection of edges). 0 The value or capacity of the cut is the total flow of the edges that are cut (the sum of the capacities (weights) of the edges).

0 The minimum cut is the cut with the minimum value. 0 The maximum flow through a network is equal to the value of the minimum cut. 0 Maximum flow = Capacity of Minimum cut

0 In some networks it is possible to produce a cut in which an edge actually heads back inside the cut rather than being directed out of the cut. 0 If an edge does this, then its flow value is set to zero in the calculation of the cut value. 0 The inside of a cut is the side on which the source node lies. 0 Sometimes the inside of a cut is shaded.

Example 1: Determine the values of the cuts made on the network diagram below and find the maximum flow: Solution: Minimum cut = Maximum Flow, therefore the maximum flow is 18.

Example 2: For the network diagram below: Determine the value of all possible cuts to give the value of the minimum cut and hence the maximum flow through the network.

Solution:

Example 3: Determine the capacities of each of the cuts in the graph shown, and also the value of the minimum cut and hence the maximum flow.

Solution:

Example 4: Determine the maximum flow for the directed graph. Solution: Cut includes segment AT, AB, BC, CT. Ignore AB as it heads back. Cut = Maximum flow = 7