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The Travelling Salesman The Travelling Salesman ProblemProblem
(TSP)(TSP)
H.P. WilliamsProfessor of Operational Research
London School of Economics
22
A Salesman wishes to travel around a given set of cities, and return to the beginning, covering the smallest total distance
Easy to State
Difficult to Solve
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If there is no condition to return to the beginning. It can still be regarded as a TSP.
Suppose we wish to go from A to B visiting all cities.
AB
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If there is no condition to return to the beginning. It can still be regarded as a TSP.
Connect A and B to a ‘dummy’ city at zero distance(If no stipulation of start and finish cities connect all to dummy at zero distance)
AB
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If there is no condition to return to the beginning. It can still be regarded as a TSP.
Create a TSP Tour around all cities
AB
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A route returning to the beginning is known as a Hamiltonian Circuit
A route not returning to the beginning is known as a Hamiltonian Path
Essentially the same class of problem
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Applications of the TSPRouting around Cities
Computer Wiring - connecting together computer components using minimum
wire length
Archaeological Seriation - ordering sites in time
Genome Sequencing - arranging DNA fragments in
sequence
Job Sequencing - sequencing jobs in order to minimise total set-up time between jobs
Wallpapering to Minimise Waste
NB: First three applications generally symmetricLast three asymmetric
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8am-10am
2pm-3pm
3am-5am7am-8am10am-1pm
4pm-7pm
Major Practical Extension of the TSP
Vehicle Routing - Meet customers demands within given time windows using lorries of limited capacity
Depot
6am-9am
6pm-7pm
Much more difficult than TSP
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History of TSP
1800’s Irish Mathematician, Sir William Rowan Hamilton
1930’s Studied by Mathematicians Menger, Whitney, Flood etc.
1954 Dantzig, Fulkerson, Johnson, 49 cities (capitals of USA states) problem solved
1971 64 Cities1975 100 Cities
1977 120 Cities
1980 318 Cities
1987 666 Cities
1987 2392 Cities (Electronic Wiring Example)
1994 7397 Cities
1998 13509 Cities (all towns in the USA with population > 500)
2001 15112 Cities (towns in Germany)
2004 24978 Cities (places in Sweden)
But many smaller instances not yet solved (to proven optimality)But there are still many smaller instances which have not been solved.
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Recent TSP Problems and Recent TSP Problems and Optimal Solutions Optimal Solutions
fromfrom
Web Page of William Cook, Web Page of William Cook, Georgia Tech, USAGeorgia Tech, USA
with Thankswith Thanks
1111
TTSSPP HHiissttoorryy
>> TTSSPP iinn PPiiccttuurreess
>> 22000044:: 2244997788
Printed Circuit Board 2392 cities 1987 Padberg and Rinaldi
1954n=49
1962n=33
1977n=1201987n=5321987n=6661987n=23921994n=73971998n=13509n=1512
2004n=24978
1212
USA Towns of 500 or more population USA Towns of 500 or more population 13509 cities 1998 Applegate, Bixby, 13509 cities 1998 Applegate, Bixby,
ChvChvátal and Cookátal and Cook
1313
Towns in Germany 15112 Cities Towns in Germany 15112 Cities
2001Applegate, Bixby, Chv2001Applegate, Bixby, Chvátal and Cookátal and Cook
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Sweden 24978 Cities 2004 Applegate, Bixby, ChvSweden 24978 Cities 2004 Applegate, Bixby, Chvátal, Cook and átal, Cook and HelsgaunHelsgaun
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Solution Methods
I. Try every possibility (n-1)! possibilities – grows faster than exponentially
If it took 1 microsecond to calculate each possibility takes 10140 centuries to calculate all possibilities when n = 100
II. Optimising Methods obtain guaranteed optimal solution, but can take a very, very, long time
III. Heuristic Methods obtain ‘good’ solutions ‘quickly’ by intuitive methods.
No guarantee of optimality
(Place problem in newspaper with cash prize)
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The Nearest Neighbour Method (Heuristic) – A ‘Greedy’ Method
1. Start Anywhere
2. Go to Nearest Unvisited City
3. Continue until all Cities visited
4. Return to Beginning
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A 42-City Problem The Nearest Neighbour Method (Starting at City 1)
1
21
20
19
29
7
30
28
3137
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
1523
22
213
16
317 4
18
42
40
41
5
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The Nearest Neighbour Method (Starting at City 1)
1
21
20
19
29
7
30
28
3137
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
1523
22
213
16
317 4
18
42
40
41
5
1919
The Nearest Neighbour Method (Starting at City 1) Length 1498
1
21
20
19
29
7
30
28
3137
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
1523
22
213
16
317 4
18
42
40
41
5
2020
29
Remove Crossovers
1
21
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7
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28
31 37
32 36
11
9
34
10
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35
12
33
8
2726
624
25
14
15 23
22
213
16
317 4
18
42
40
41
5
2121
Remove Crossovers
1
21
20
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7
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28
31 37
32 36
11
9
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10
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35
12
33
8
2726
624
25
14
15 23
22
213
16
317 4
18
42
40
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5
2222
Remove Crossovers Length 1453
1
21
20
19
29
7
30
28
31 37
32 36
11
9
34
10
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35
12
33
8
2726
624
25
14
15 23
22
213
16
317 4
18
42
40
41
5
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Christofides Method (Heuristic)
1. Create Minimum Cost Spanning Tree (Greedy Algorithm)
2. ‘Match’ Odd Degree Nodes
3. Create an Eulerian Tour - Short circuitcities
revisited
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1
10
20
21
35
12
19
29
7 33
34
39
26
38
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25
27
246 28
2215
42
16
417
18
8
30
11
32
9
36
3731
3
40
14
13
41
2
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Christofides Method 42 – City Problem
Minimum Cost Spanning Tree
Length 1124
2525
Minimum Cost Spanning Tree by Greedy Algorithm
Match Odd Degree Nodes
2626
1
10
20
21
35
12
19
29
7 33
34
39
26
38
23
25
27
246 28
2215
42
16
417
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8
30
11
32
9
36
3731
3
40
14
13
41
2
5
1
10
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21
35
12
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7 33
34
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246 28
2215
42
16
417
18
8
30
11
32
9
36
3731
3
40
14
13
41
2
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Match Odd Degree Nodes in Cheapest Way – Matching Problem
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1. Create Minimum Cost Spanning Tree (Greedy Algorithm)
2. ‘Match’ Odd Degree Nodes
3. Create an Eulerian Tour - Short circuitcities
revisited
2828
Create a Eulerian Tour – Short Circuiting Cities revisited
Length 1436
1
21
20
19
29
7
30
28
31 37
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
1523
22
213
16
317 4
18
42
40
41
5
2929
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Optimising Method
1.Make sure every city visited once and left once – in cheapest way (Easy)
-The Assignment Problem - Results in subtours Length 1098
1
21
20
19
29
7
30
28
31 37
32 36
11
9
34
10
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12
33
8
266
24
25
14
1523
22
213
16
317 4
18
42
40
41
5
3030
Put in extra constraints to remove subtours (More Difficult)
Results in new subtours Length 1154
1
21
20
19
29
7
3028
31 37
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
15 23
22
213
16
317 4
18
42
40
41
5
3131
Remove new subtours
Results in further subtours Length 1179
1
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20
19
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7
30
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31 37
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
1523
22
213
16
317 4
18
42
40
41
5
3232
Further subtours Length 1189
1
21
20
19
29
7
30
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31 37
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
1523
22
213
16
317 4
18
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40
41
5
3333
Further subtours Length 1192
1
21
20
19
29
7
30
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31 37
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
1523
22
213
16
317 4
18
42
40
41
5
3434
Further subtours Length 1193
1
21
20
19
29
7
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31 37
32 36
11
9
34
10
39
38
35
12
33
8
2726
624
25
14
15 23
22
213
16
317 4
18
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40
41
5
3535
Optimal Solution Length 1194
1
21
20
19
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7
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31 37
32 36
11
9
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10
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12
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2726
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1523
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5
![Travelling Salesman Problemweb.tecnico.ulisboa.pt/.../legacy/PRINT/NotebookSC_5TSP_06pp.pdf · Travelling Salesman Problem [:6] 3 This is, however, not a solution to the TSP, because](https://img.dokumen.tips/doc/110x75/5fab2a9ca11f0c17d66c3999/travelling-salesman-travelling-salesman-problem-6-3-this-is-however-not-a-solution.jpg)
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