3. TRAVELLING SALESMAN
PROBLEM
Find the shortest possible route
that visits each city exactly once
and returns to the origin city
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4. P, NP, NP-COMPLETE, NP-HARD
Nondeterministic-
Polynomial Time Polynomial Time NP-Complete NP-Hard
5. P (POLYNOMIAL TIME)
» P is the set of all decision problems which can be solved in polynomial
time by a deterministic Turing machine. Since it can be solved in
polynomial time, it can also be verified in polynomial time
» E.g:
• Linear Programming -> determining a way to achieve the best
outcome (such as maximum profit or lowest cost) in a given
mathematical model
• finding Maximum Matching -> graph matching
6. NP (NON-DETERMINISTIC POLYNOMIAL)
» NP is the set of all decision problems (question with yes-or-no answer)
for which the 'yes'-answers can be verified in polynomial time (O(nk)
where n is the problem size, and k is a constant) by a deterministic
Turing machine. Polynomial time is sometimes used as the definition of
fast or quickly
» P is a subset of NP
» E.g:
• TSP
7. NP-COMPLETE
» A problem x that is in NP is also in NP-Complete if and only if every
other problem in NP can be quickly (ie. in polynomial time)
transformed into x. In other words:
• x is in NP, and
• Every problem in NP is reducible to x
» So what makes NP-Complete so interesting is that if any one of the NP-
Complete problems was to be solved quickly then all NP problems can
be solved quickly
» E.g:
• TSP
8. NP-HARD
» NP-Hard are problems that are at least as hard as the hardest problems
in NP. Note that NP-Complete problems are also NP-hard. However not
all NP-hard problems are NP (or even a decision problem), despite
having 'NP' as a prefix. That is the NP in NP-hard does not mean 'non-
deterministic polynomial time’
» E.g:
• TSP
10. TSP IS NP-HARD
U$ 1m
IF P = NP IS SOLVED
Millenium Prize
Problem
AND CREDITS FROM
SCIENTISTS AROUND
THE WORLD
source: http://www.claymath.org/millennium/P_vs_NP/
11. TSP HISTORY
» 1920: Karl Menger introduced the concept to colleagues in Vienna
» 1930: Intensive discussion in math community in Princeton University
» 1940: Merrill Meeks Flood publicized TSP to mass
» 1948: Flood presented TSP to RAND Corp. RAND is a non-profit
organization that focuses in intellectual research and development
within the US
» 1950: Linear Programming was becoming a vital force in computing
solutions to combinatorial optimization problems. The US Airforce
needed the method to optimize solutions of their combinatorial
transportation problem
» 1960’s: The TSP could not be solved in polynomial time using Linear
Programming techniques
12. TSP has never been
solved
by scientists and experts so far
13. TSP OVERVIEW (1)
» Find the shortest possible route that visits each city exactly once and
returns to the origin city -> Hamiltonian cycle
» Posed such computational complexity that any programmable efforts
to solve such problems would grow superpolynomially with the
problem size
» Can be used in :
• transportation: school bus routes, service calls, delivering meals
• manufacturing: an industrial robot that drills holes in printed
circuit boards
• VLSI (microchip) layout
• communication: planning new telecommunication networks
14. TSP OVERVIEW (2)
» One way to solve TSP is to use exhaustive search to find all possible
combinations of the next city to visit
» However, the method is costly, since the number of possible tours of a map
with n cities is (n − 1)! / 2
#cities #tours
5 12
6 60
7 360
8 2,520
9 20,160
10 181,440
» 25 cities will require:
310,224,200,866,619,719,680,000
15. TSP OVERVIEW (3)
Vehicle Routing - Meet customers demands within given time windows
using lorries of limited capacity
10am-1pm 7am-8am 3am-5am
4pm-7pm 6pm-7pm
Depot
8am-10am
6am-9am
2pm-3pm
Much more difficult than TSP
16. TSP OVERVIEW (4)
» Until this very day, an efficient solution to the general case TSP, or even
to any of its NP-hard variations, has not been found
» However, there are approximation solutions to solve the TSP:
• Polynomial Time Approximation Scheme (PTAS)
• Christofides Algorithm
• Double MST Algorithm
• Arora’s Algorithm
• Mitchell’s Algorithm
