1. Traveling Salesperson Problem

2. - 3 4 6 2 5 7 Traveling Salesperson Problem (TSP) A B A B C D A B C D
Optimization Problem: Find the least cost route starting
at A, traveling through the other three cities exactly once and
returning to A.
Decision Problem: Is there a TSP route with cost less than k?

3. Icosian Game 1857

4. Traveller’s Dodecahedron 1857

5. Sir William Hamilton 1805 - 1865 Discovered quaternions in 1843
Invented Icosian Game in 1857
Invented Traveller’s Dodecahedron
Hamiltonian Circuits

6. T. P. Kirkman In 1856, he published sufficient
In 1856, he published sufficient
conditions for a polyhedral
graph to have a Hamiltonian
circuit.

8. Twentieth Century History
Merrill Flood, A.W.Tucker, Hassler Whitney (Princeton)
pose the Traveling Salesperson Problem.
1947 George B. Dantzig designs simplex method for linear programming
problems and formalizes combinatorial optimization as a
a field of study.
1954 Connection with the Assignment Problem established.
1962 Dynamic programming formalized.
1972 Karp proved TSP is NP-complete.
1985 – 2006 Techniques leading to solving 300-city problems to …

9. TSP Applications
Laser drilling circuit boards
Robotic welding of cars

10. Genome sequencing. Finding most likely transitions between local maps.
TSP Applications
Genome sequencing. Finding most likely transitions between local maps.
Cities = maps
Cost = likelihood of transition
Job processing: Chemical plants – cost of setup to produce chemicals.
Cities = setups
Cost = time and cost of moving from one setup to next.
Vehicle Routing Problem: Central depot delivery.
Cities = Locations
Cost = Delivery cost (minus profit)
Depot

11. Admissions Office Campus Tour
Mathais
Herb ‘n Farm
Slocum
El Pomar
Packard
Admissions Office Campus Tour

12. A
A B C D B - 3 4 6 2 5 7 A B C D C D
Traveling Salesperson Problem: Find the least cost route
starting at A, traveling through the other three cities exactly
once and returning to A.

13. Naïve Algorithm
1. List all routes and their costs:
A BCD A = 18
A BDC A = 19
A CBD A = 15
A CDB A = 15
A DBC A = 24
A DCB A = 19
2. Find a route with minimum cost:
A CBD A = (one optimal route)

14. Complexity
If there are n cities (in addition to the home city A), then
there are n! possible routes.
The naïve algorithm takes at least n! steps.
A random algorithm could possibly find an optimal
route in n steps.
Is there a deterministic algorithm that can find an optimal
route in a polynomial number of steps?
Finding the minimum in a list takes about n steps and
sorting a list takes about n2 steps (or nlogn if you are
clever.)

15. Theorem: (1972) TSP is NP-Complete. (That is, every other
hard problem is reducible to TSP.)
Open problem: Is there a polynomial time algorithm for TSP?
If not, can you prove it? (In technical terms, does P = NP?)
Solving the open problem earns you $1,000,000 from the Clay
Mathematics Institute.

16. Generating Permutations:
Recursion:
A < BCD B < ACD C < ABD D < ABC
BDC ADC ADB ACB
CBD CAD BAD BAC
CDB CDA BDA BCA
DBC DAC DAB CAB
DCB DCA DBA CBA
Single exchange:
ABCD CABD DCBA DBAC
BACD ACBD DCAB DBCA
BCAD ACDB DACB BDCA
BCDA CADB ADCB BDAC
CBDA CDAB ADBC BADC
CBAD CDBA DABC ABDC

17. Timing results for Naïve Algorithm:
Cities Seconds Routes Sec/Route 10
0.50
3,628,800
1.38 x 10-7 11
5.61
39,916,800
1.41 x 10-7 12
69.12
479,001,600
1.44 x 10-7
Estimate for 16 cities: days (2.09 x 1013 routes)

18. Total nodes: 1+ n + n(n-1) + … + n!B C D C D B D B C
Little work
required. D C D B C B
Total nodes: 1+ n + n(n-1) + … + n!
(If n=3, there are 10 nodes.)

19. Dynamic Programming Algorithm:
A – BCD EFG – A
A – BCD EGF – A
A – BCD FGE – A
A – BCD FEG – A
A – BCD GFE – A
A – BCD GEF – A
A – BCD FEG – A
A – BDC FEG – A
A – DBC FEG – A
A – DCB FEG – A
A – CBD FEG – A
A – CDB FEG – A
FEG
g(i, S) = shortest path from i through vertices in S ending at A.
g(i, S) = min { d g(j, S - {j}) }
Optimal route = g(A, S) Steps: n2 x 2n
i,j
j in S

20. A B C D C D B C B D C C
Branch and Bound techniques possibly reduce the number of
nodes visited.

21. Branch and Bound Technique
Select node A.
Generate children of selected node.
If children complete a route, update optimal route.
Otherwise, estimate cost of each child node and put on possible list.
If estimated cost is greater than optimal so far, remove from list.
Remove currently selected node from possible list.
Select node on possible list with smallest cost.
If there is no selected nodes, STOP, else go to 2.
Technique depends on how accurate estimate is. Estimate must be a
lower bound on the cost and is called the bounding function.

22. A Possible Bounding Function B(i) :
Each node i in the search tree represents a partial route. Those
cities not in the partial route form a set of “remaining” cities.
C1 = cost of partial route.
C2 = sum of the minimum edges into each remaining city.
B(i) = C1 + C2
B(i) <= Cost of any route containing the partial route.

23. Timing results for Branch and Bound Algorithm:
(Averages for three instances.)
Cities Seconds Nodes Total Nodes 10
0.003
308
6,235,301 11
0.008
1442
68,588,312 12
0.011
3222
823,059,745 20
1.667
845,035
5 x 1019
For 22 cities, the elapsed time is about 17 seconds.
Remember that the time and nodes visited vary with the TSP instance.

24. Shortest Campus Tour: 1281 seconds
Mathais
Shortest Campus Tour: seconds
Herb ‘n Farm
Tutt Library
Cutler
Slocum
El Pomar
Armstrong
Elapsed time: 1.3 seconds
Total nodes visited: 743, 237

25. Linear Programming formulation of TSP

26. x2
x1+ x2 = 5.8
Feasible region
Minimize: x1+ x2
Subject to: 2x1 + x2 >= 5
x1 + 2x2 >= 6
x1 >= 0
x2 >= 0 x1
x1+ x2 = 1
x1+ x2 = 3.67

27. TSP Instances Asymmetric Symmetric Upper Triangular Euclidean
Constant cost

28. - 1 3 5 10 6 8 2 9 A A B C D B A B C D C D Constant TSP:
A BCD A = 22 A CBD A = 22 A DBC A = 22
A BDC A = 22 A CDB A = 22 A DCB A = 22

29. Constant TSP: a b c 1 4 2 3 6 2 4 - 1 3 5 10 6 8 2 9

31. APPROXIMATIONS
Are there “rules” (called heuristics) that help find good, if not
optimal, TSP routes?
Could an approximation algorithm find a route that was, say,
within 110% of the optimal route?
If I is a particular (instance) TSP problem, and A is an
approximation algorithm, let A(I) be the cost of the route
it generates and let OPT(I) be the optimal cost. Is it possible
That A(I) < 1.1 OPT(I) for all I?
Bottom line: No polynomial time algorithm A can guarantee
that for all instances I of TSP, A(I) < r OPT(I) for any
constant r, unless P=NP.

32. Geometric version of TSP:
The triangle inequality holds for the distance matrix.
For any three cities A, B, C,
AB + BC >= AC
If the distance is actually the normal Euclidean distance,
then the instance is referred to as a Euclidean TSP.
For the Euclidean TSP, the route does not cross itself.
The Euclidean TSP is still NP-Complete, but there are
approximation algorithms.

33. Edge Crossings in Geometric TSPA B e D C
Ae + eB >= AB and Ce + eD >= CD
AD + CB >= AB + CD

34. Shortest Campus Tour: 1281 seconds
Mathais
Shortest Campus Tour: seconds
Herb ‘n Farm
Tutt Library
Cutler
Slocum
El Pomar
Armstrong
Elapsed time: 1.3 seconds
Total nodes visited: 743, 237

35. Minimum Spanning Tree Algorithm:
Pick next smallest edge connected to partial tree
but not forming a cycle. D 11 C 6 9 E 7 2 4 7 10 9 A 5 B

36. Minimum Spanning Tree Algorithm:
Pick next smallest edge connected to partial tree
but not forming a cycle. D 11 C 6 9 E 7 2 4 7 10 9 A 5 B

37. Minimum Spanning Tree Algorithm:
Pick next smallest edge connected to partial tree
but not forming a cycle. D 11 C 6 9 E 7 2 4 7 10 9 A 5 B

38. Minimum Spanning Tree Algorithm:
Pick next smallest edge connected to partial tree
but not forming a cycle. D 11 C 6 9 E 7 2 4 7 10 9 A 5 B

39. Minimum Spanning Tree Algorithm:
Pick next smallest edge connected to partial tree
but not forming a cycle. D 11 C 6 9 E 7 2 4 7 10 9 A 5 B

40. Minimum Spanning Tree Algorithm:
Pick next smallest edge connected to partial tree
but not forming a cycle. D 11 C 6 9 E 7 2 4 7 10 9 A 5 B
To form a TSP route, trace the minimum spanning tree twice using short cuts if possible. This gives A-E-B-C-D-A.
The cost of this route is 31. The spanning tree has cost 19.

41. Let OPT = cost of optimal TSP route.
MST = cost of minimum spanning tree.
Rmst = cost of route derived from minimum spanning tree.
The optimal TSP route minus the last edge is a
minimum spanning tree, so MST < OPT.
Then Rmst <= 2*MST <= 2*OPT
With a little more care the shortcuts can be optimized to give:
Rmst <= 1.5*OPT
Recently an algorithm was discovered that gives
Rmst <= (1+e)*OPT for arbitrary e>0

42. APPROXIMATIONS
Some Heuristics for finding a good TSP tour:
Nearest Neighbor
Dissection
Nearest Insertion
Tour improvement (Lin-Kernighan)

43. Ant Colony Optimization
Ants find shortest path to food.
They deposit pheromone on trail.
Trails with most pheromone get most ants.
Basic searching behavior has random character.

44. Ant Colony Optimization Algorithm for TSP:
Place some ants at each city.
2. Each ant calculates a = c / d where c is the amount
of pheromone and d is the distance between i and j.
Each ant selects a city not already visited with probabilities
proportional to the a .
After all ants have built a tour, evaporate some fraction
of the existing pheromone.
5. Each ant deposits an amount of pheromone inversely proportional to the length of their tour.
6. Iterate steps 1 – 4 a large number of times. r s
i,j
i,j
i,j
i,j
i,j
i,j

45. Ant Campus Tour: 1291 seconds
Mathais
Ant Campus Tour: seconds
Herb ‘n Farm
Tutt Library
Cutler
Slocum
El Pomar
Armstrong
Elapsed time: 1.6 seconds
Ant Tours: 34,000

46. Milestones in solving the TSP
Timing:
d years of CPU time (110 processors; 500mhz.)
sw years of CPU time

47. Line drawings with TSP routes: (Robert Bosch and Craig Kaplan)