Traveling
Salesman
Problem (TSP)
Chris Seto
Andrea Smith
Problem description
 Given
a list of a cities and distance
between each pair of cities, what is the
shortest possible route that visits each?
 NP-hard problem
 Problem usually modeled as an
undirected graph
 Produces a Hamiltonian Cycle
History
 Studies
started in the 1800s by Sir William
Hamilton and Thomas Kirkman of related
problems
 Icosian game invented in 1857
 TSP first studied in 1930s by Karl Menger,
Hassler Whitney, and Merrill Flood
 Solutions appeared in papers in mid-1950s
 Determined to be NP-hard in 1972 by
Richard M. Karp
Application: Pick And Place
 Printed
circuit board (PCB) with locations
where chips must be placed by robot
 The faster the robot can place all
components, the faster the PCB will be
assembled
 The faster the PCB can be assembled, the
more PCBs can be made in the same
amount of time
Application: Logistics
 Warehouse
with many parts in various
locations
 Order is received for several parts which
must be picked by robot or employee
 What path should the picker follow to
ensure that they fill the order in smallest
amount of time?
 Any number of TSP solutions can be
applied and compared in parallel
Application: UPS ORION
 Route
optimization used by UPS
 Implementation started in 2008
 Saves ~35 million miles per year
 Increased projected annual savings
NP Hard Solution Methods
 Devise
an algorithm for an exact solution,
even though it may only work efficiently
for a small problem
 Devise “Suboptimal” heuristic algorithms
to yield good, but inexact solutions.
 Find special cases for the problem for
which better or exact heuristics are
developed.
Possible Approaches
 Brute-force

Best for small number of nodes
 Greedy

Algorithm
Simplest algorithm for larger number of
nodes
 Genetic

Method
Algorithm
Generates “close to optimal” solutions
Approach: Brute Force
•
•
•
•
•
Best for small number of nodes
Number of Hamiltonian Circuits = (n-1)!
Guaranteed an Optimal Solution
Complexity: ((n-1)!)
Tries all possible permutations and
compares cost
Node Count
Approximate Completion Time
20
2 Minutes
25
20 Years
30
284 Million Years
Brute Force Demo
Approach: Greedy
•
•
•
Prim’s Algorithm, Kruskal’s Algorithm
Not guaranteed the optimal solution
“Close enough” solution
𝑛 𝑛−1
• Prim’s Complexity: O(
lg(𝑛))
2
𝑛 𝑛−1
𝑛 𝑛−1
• Kruskal’s Complexity: O(
lg(
2
2
))
Approach: Genetic Algorithm
 New
approach which uses natural
selection to create close to optimal
solutions
 Hamiltonian cycles are continually “bred”
with mutations
 Crossover occurs between solutions
 Relatively quickly produces a solution
which is probably close to optimal
Conclusion



Studies originally began 1800s, again in 1950s
Optimal solution found in ((n-1)!) time
Close enough solution found in
O(

𝑛 𝑛−1
2
lg(𝑛))
Used by…



UPS in ORION system
Pick and place
Many other graph representable systems
Questions?
References




http://en.wikipedia.org/wiki/Travelling_salesm
an_problem
https://xkcd.com/399/ (Comic)
http://www.forbes.com/sites/alexkonrad/2013
/11/01/meet-orion-software-that-will-save-upsmillions-by-improving-drivers-routes/
http://www.theprojectspot.com/tutorialpost/applying-a-genetic-algorithm-to-thetravelling-salesman-problem/5