1. A computational study & tool for the Travelling Salesman Problem (TSP)
Mangesh Gharote, Dilys Thomas and Sachin Lodha
Tata Consultancy Service – Innovations Lab
Note: A general presentation for research and study purpose.
Original Article found on
Cite as: Mangesh Gharote, Dilys Thomas, Sachin Lodha:
Excel Solvers for the Traveling Salesman Problem. COMAD 2012:

2. Talk outline Problem statement Applications Computational Complexity
Top Awards
Combinatorial algorithm
nearest neighbour
greedy
Geometric repair heuristics
intersection removal
hinge and crest optimization
Benchmark datasets
cities, PCB layout etc.
comparison of algorithms to state of art

3. Problem Statement, Complexity
Given
n cities,
Distances or costs between pair
or a metric on the space
Find
Permutation such that tour length is minimized
Problem
There are n! different permutations
An actual tour on the globe with
the lines indicating tour paths
n=30 beyond reach of all computers on earth combined
100’s of mathematicians and computer scientist have tried to solve this problem

4. Applications BPO Vehicle Routing Bus Scheduling
Development of flight schedules
Crew Scheduling
Order-picking problem in warehouses
Printing press scheduling problem
Network cabling in a country
Computer wiring
Query workload ordering for optimization
VLSI chip design connectivity layout
Drilling of printed circuit boards

5. Applications Hot rolling scheduling problem in Iron & Steel Industry
Overhauling gas turbine engines (Ordering nozzle guides)
X-Ray crystallography (Ordering positions for measurement)
Global navigation satellite system
Ordering test cases in regression suite to re-use components

6. Popular Appeal
TSP in 1960 achieved national prominence in the United States of America
Procter & Gamble offered prizes up to $10,000 for identifying the most correct links in a particular 33-city problem.

7. Awards Proved to be NP complete in 1972 by Richard Karp
Richard Karp won the Turing Award 1985
For contribution to the theory of algorithms and NP completeness including this result
He showed a reduction from a result on logic satisfiability of conjunctive normal form clauses that have 3 literals each (3SAT)
The most recent 2010 Gödel Prize
Awarded to Sanjeev Arora and Joseph S B Mitchell for polynomial time approximation scheme for Euclidian TSP

8. Objective of our work
Develop our own algorithms on top of reasonable in practice TSP algorithms
Near optimal in practice
Reduce run time for medium to large instances
Scalable in memory for medium to large instances
Ease of using the tool, handle different distance metrics including longitude and latitude
Easy to visualize tours
Improve state of art TSP solvers available in Excel

9. Different Approaches for Solving TSP
Exact
Enumeration
Linear Programming
- Miller–Tucker–Zemlin (1960)
Dynamic Programming
Approximation
Algorithms
Nearest Neighbour
Greedy
K-Opt, Kernighan Lin (70’s)
Christofides (1965)
Heuristics
Simulated Annealing
Genetic Algorithm
Ant Colony Optimization
Electromagnetism like Algorithm
Lower Bound
Held -Karp

10. Approximation Algorithms I) Nearest Neighbour II) Greedy Algorithm

11. Nearest Neighbour (NN) Algorithm for TSP (Pseudo code)
1. Select an arbitrary vertex as current vertex.
2. Find out the shortest edge connecting current vertex and an unvisited vertex V.
3. Set current vertex to V.
4. Mark V as visited.
5. If all the vertices in domain are visited, then terminate; otherwise, go to step 2.
Time Complexity:
O(n2)

12. Greedy Algorithm for TSP (Pseudo code)
1. Sort all edges in increasing order of weigths.
2. Select the shortest edge and add it to our tour if it doesn’t violate any of the below constraints:
if it is not yet in the tour
if adding it would not create a degree-3 vertex or
If does not create a cycle of size less than N.
3. Do we have N edges in our tour? If no, repeat step 2.
Time Complexity
O( n2 log (n))
Sub Cycle
Degree of Vertex < 3

13. Numerical Example Using NN & Greedy1 2 3 4
Input Graph 4 4 1 2 1 2 3 2 4 2 3 4 3 4 2 2
Starting Node 12 11
Greedy
Nearest Neighbour

14. Nearest Neighbour with different starting vertex 16 Cities Benchmark Problem
Difference from
Optimal 5 %
Start
Start
Difference from
Optimal 32 %

15. 16 Cities Benchmark Problem
Greedy
Difference from
Optimal 16 %

16. 51 Cities Benchmark Problem
Nearest Neighbour
Greedy
Difference from Optimal
18.7 %
Difference from Optimal
13 %

17. Results on Benchmark Problems Using Nearest Neighbour (NN) and Greedy

18. Observation about NN and Greedy
For ‘N’ node graph, theoretically can guarantee only upto O(log N) multiplicative factor away from optimal
towards the end more expensive edges could be forced
Start point makes a difference for nearest neighbour
In practice gives 25% away from optimal for moderately large sized instances
They can be improved further with geometric constructions
2-opt, intersection removal, hinge-crest optimization

19. Types of Distances 2D – Euclidean ND – Euclidean (L1 , L2, Linfty)
Geographical (Longitude – Latitude)
Symmetric - Asymmetric
Triangular Inequality

20. Geographical Distance Code - Longitude & Latitude
PI = For computing the geographical distance; RRR = ; /* Radius of earth in kms */ deg = (int) X[i]; min = X[i] – deg; rad = PI * (deg * min/3.0)/180.0; q1 = cos (longitude[i] – longitude[j]); q2 = cos (latitude[i] – latitude[j]); q3 = cos (latitude[i] + latitude[j]); Dij = (int) (RRR * acos(0.5*((1.0 + q1)*q2 – (1.0 – q1)*q3))+1.0);

21. World Tour Airports
NN – 2 Opt
19% Away from Optimal

22. World Tour - 535 Airports 10 % Greedy 2 Opt Difference from Optimal
Alaska
North America
Europe
Hawai
Guam
Asia
Africa
Fiji
South America
Australia
Santiago
Trans Pacific Flights

23. USA 48 Mainland Capitals

24. Africa and Islands Optimal Tour Sahara Senegal St. Helenas Mauritius
Tristan de cunha
South Africa

25. PCB- Drilling Problem (Nodes-2103)
Optimal Solution
Actually our algorithm ensures
it is not a criss-cross
Referred as one of the Nasty TSP
NN 2 Opt 6.5 %

26. State-of-the-art problem sizes
From: Book: Travelling Salesman Problem: A computational study
David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook

27. State-of-the-art optimal solutions
From: Book: Travelling Salesman Problem: A computational study
David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook

28. State-of-the-art Concorde Solver
From: Book: Travelling Salesman Problem: A computational study
David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook

29. Largest TSP instances solved optimally till date
Ref:

30. Motivating Our Work: Need for Approximate Solutions
Year – 2001
15,112 German towns solved using 110 processors at Rice University and Princeton University ~ 22.6 years on 500MHz Alpha processor.
Year – 2005
33,810 Circuit board using state-of-the-art Concorde TSP Solver took 15.7 CPU - years
Year -2006
85,900 points was solved using Concorde TSP Solver took over 136 CPU-years.

31. Our Benchmark Results

32. Well Known Approximation Algorithms
Lin-Kernighan
heuristic works well in practice
Linear Programming Formulation for TSP
basis for using CPEX, Gurobi, Xpress solvers
Held-Karp Dynamic Programming (n2 2n)
40 year open problem if there is an exact algorithm for TSP with time (cn) for c < 2.
Christofides 1.5 Approximation
30 year open problem if there is an approximation algorithm with factor < 1.5

33. Lin-Kernighan Costly to find 3-opt improvements: O(n3) candidates
k-opt: generalizes 3-opt O(nk)
Tour modification:
Collection of simple changes
Some increase length
Total set of changes decreases length

34. Linear Programming Formulation for TSP
Degree of Vertex is 2
Subtour elimination

35. Held-Karp Dynamic Programming
O(n22n) algorithm for TSP
Uses Dynamic programming
Take some starting vertex s
For set of vertices R (s Î R), vertex w Î R, let
B(R,w) = minimum length of a path, that
Starts in s
Visits all vertices in R (and no other vertices)
Ends in w
A&N: TSP

36. TSP: Recursive Dynamic Programming formulation
B({s},s) = 0
If |S| > 1, then
B(S,w) = minv Î S – {w}B(S-{w}, v}) + weight(v,w)
If we have all B(V,v) then we can solve TSP.
Gives requested algorithm using DP-techniques.
A&N: TSP

37. Christofides 1.5 Approximation
Make a Minimum Spanning Tree T
Set W = {v | v has odd degree in tree T}
Compute a minimum weight matching M in the graph G[W].
Look at the graph T+M. (Note: Eulerian!)
Compute an Euler tour C’ in T+M.
Add shortcuts to C’ to get a TSP-tour
A&N: TSP

38. Work done by students (2010-11)
Metaheuristics:
Simulated Annealing
Ant Colony Optimization
Electromagnetism-like Algorithm
Project Title:
“Excelling with Metaheuristics”
B.E. Project Done By:
Prem Nathan, Prashant Kumar and Sani Kumbhar

39. Simulated Annealing (Kirkpatrick, 1985)
SA is a variant of local (neighborhood) search
Traditional local search always moves in a direction of improvement
SA allows non-improving moves to avoid getting stuck at a local optimum
Analogy is from Material Science
Slowly cool down a heated solid, so that all particles arrange in the ground energy state
At each temperature wait until the solid reaches its thermal equilibrium
Probability of being in a state with energy ‘E’ given by Boltzmann

40. Ant Colony Optimization (Dorigo et al 1996)
Ants secrete pheromone while traveling, in order to communicate with one another to find the shortest path
The more ants follow a trail, the more attractive that trail becomes for being followed.
Even when the tracks are equal the behavior will encourage one over the other—convergence

41. Electromagnetism Like Algorithm (Birbil & Fang, 2003)
Initialization
Force Calculation F
Movement
Neighborhood
Search
Idea is based on the attraction repulsion mechanism of electromagnetism theory (Coulomb’s law)
force exerted on a point via other points is inversely proportional to the distance between the points and directly proportional to the product of their charges
The charge of each particle relates to the value of objective function
the better the objective function value, the higher the magnitude of attraction.

42. Results – Metaheuristics
Nodes
EM (%)
ANT(%)
SA (%) 14
5.3
3.0
7.6 52
7.1
6.4
18.2 96
9.9
6.0
15.2
159
15.4
14.3
20.5
226
15.9
13.1
11.8
299
22.9
18.5
24.7
654
24.2
24.9
25.3
Nodes EM
Time
(min) SA
1291
14.6%
3.1
18.0%
18.2
2103
6.3%
12.1
8.7%
34.3
3795
13.1%
20.1
19.0%
24.3
4461
20.0%
6.1
26.2%
31.1
5934
18.7%
23.8%
68.1
14051
21.4%
16.5
22.9%
44.6

43. Summary TSP - Geometric NP hard problem
Exact solution expensive (time, space)
Implemented combinatorial algorithms
NN, Greedy, intersection removal, hinge-crest optimization
Provide approximations in practice
Can solve and visualize tours of medium to large size
(e.g. 14,000 cities – two orders of magnitude larger than Elsevier paper)
Multiple applications to PCB layout, X-ray crystallography, other engineering and scientific domains
Future Plans – To improve on solving even larger size problems by optimizing algorithm implementations
Privacy Preserving TSP

44. Demonstration – TSP Using Excel
48 US Mainland Capitals
NN, NN 2-Opt, Greedy, Greedy 2-Opt
Demo 2
Printed Circuit Board Drilling Problem
280 Locations on 2D plane
NN and NN-2 Opt
All intersections removed

45. Thank You ! Reference Robert Bosch, February 2009 mona-lisa100K.tsp
$1000 Prize Offered
[Can convert pictures into tours using Voronoi diagrams]