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Math 103, Section 11, Friday, April 15, 2011
Homework for Chapter 5 is due on Friday, April 15, 5 minutes before midnight.
Please do Exercises (on pages 193196)
20,26,30,34,38,42 and 46 and upload
to Sakai.
Homework for Chapter 6 will be discussed in class and not submitted to Sakai.
Exercises (on pages 229232) 24,26,30,32,40,46
There will be a quiz on Chapter 6 on Tuesday, April 26.
Homework for Friday, April 15. Page 229 Exercises 23,25,29
Chapter 6 The Mathematics of Touring
Hamilton Paths and Hamilton Circuits and The Traveling Salesman Problem
Graph model of a traveling salesman problem
Sites
vertices of the graph
Costs
weights of the edges
Tour
Hamilton circuit
Optimal tour
Hamilton circuit of least total weight
A
tour
is a Hamilton circuit of the graph and an
optimal tour
is the Hamilton circuit of
least total weight.
Strategies for solving Traveling Salesman Problems (TSPs)
We will learn:
1. Exhaustive Search. Also called the BruteForce Method.
2. NearestNeighbor Algorithm
3. Repetitive NearestNeighbor Algorithm
4. The CheapestLink Algorithm
Table 2 completed
Starting
vertex
Route
Route
starting with
A
Cost of route
A
A,C,E,D,B,A
A,C,E,D,B,A
$773
B
B,C,A,E,D,B
A,E,D,B,C,A
$722
C
C,A,E,D,B,C
A,E,D,B,C,A
$722
D
D,B,C,A,E,D
A,E,D,B,C,A
$722
E
E,C,A,D,B,E
A,D,B,E,C,A
$741
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View Full Document Note that trips starting at vertices B,C, and D are the same trip . You can see this when
you rewrite the trip so that the Route starts at vertex A.
So the repetitive nearest neighbor algorithm gives the tour A,E,D,B,C,A
with $722 as
the cheapest.
Relative error of a tour (e)
e=(cost of tourcost of optimal tour)/cost of optimal tour
relative error for the repetitive nearest neighbor algorithm=(722676)/676=
0.068047=6.80%
Exercises:
Brute force and nearest neighbor exercises.
Solution to Exercise 29:
Part a:
You can use the following table to find the optimal tour:
Hamilton
circuit
Total cost
1
A,B,C,D,A
48+32+18+22=120
2
A,B,D,C,A
48+20+18+28=114
3
A,C,B,D,A
28+32+20+22=102
optimal tour
4
A,C,D,B,A
28+18+20+48=114
5
A,D,B,C,A
22+20+32+28=102
another optimal tour
6
A,D,C,B,A
22+18+32+48=120
b.f.
The nearestneighbor tours with costs, rewritten with starting vertex A:
Starting
vertex
Route
Route
starting with
A
Cost of route
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This note was uploaded on 01/20/2012 for the course MATH 103 taught by Professor Berkowitz during the Spring '07 term at Rutgers.
 Spring '07
 Berkowitz
 Math

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