Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.2, Slide 1 4 Graph Theory (Networks)...

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Section 4.2, Slide 1

4 Graph Theory (Networks)

The Mathematics of Relationships

4



The Traveling Salesperson Problem

4.2

• Understand how to solve the traveling salesperson problem using Hamilton circuits

• Determine all Hamilton circuits in a complete graph

(continued on next slide)




4.2

• Solve the traveling salesperson problem using the brute force

algorithm

• Solve the traveling salesperson problem using the best edge

algorithm

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.2, Slide 4


Danielle lives in Philadelphia and must make visits next week to branch offices in New York City, Cleveland, Atlanta, and Memphis. To determine which would be her cheapest trip, she has obtained prices of flights between each pair of cities.

What sequence of visits minimizes the cost?


Hamilton Paths

• Before we tackle Danielle’s problem, we need some additional tools.


Hamilton Paths

• Example: Find a Hamiltonian path.

(solution on next slide)


Hamilton Paths

• Solution: answers may vary.


Finding Hamilton Circuits

• Examples:



• Often, we need to find all the Hamilton circuits in a graph.

• This is easy for complete graphs.

• Tree diagrams help us find Hamilton circuits systematically.



• Example: Find all Hamilton circuits in K4.

• Solution:Path ABCDA is a Hamilton circuit in K4.

We will consider path BCDAB to be the same path because it passes through the same vertices in the same order and the only difference is that we are beginning and ending at vertex B rather than vertex A.

So, we will assume that all circuits begin at A.




• Solution:

Use a tree diagram to list all Hamilton circuits systematically.




• Solution:


Solving the TSP by Brute Force

• In solving a TSP problem by brute force, we consider all possible Hamiltonian circuits.



• Example: Use the weighted graph to find the sequence of cities for Danielle to visit that will minimize her total travel cost.



• Solution: Use brute force to explore all possible circuits:


The Nearest Neighbor Algorithm

• There are algorithms that give good approximations to the TSP.


The Nearest Neighbor Algorithm

• Example: Use the nearest neighbor algorithm to schedule Danielle’s trip.

• Solution:


The Best Edge Algorithm


The Best Edge Algorithm

• Example: Use the best edge algorithm to schedule Danielle’s trip.

• Solution:

210 + 230 + 170 + 240 + 350 = 1200Notice that this circuit has a weight of 1,200, which also makes it the best solution to Danielle’s problem.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.2, Slide 1 4 Graph Theory (Networks)...

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