Raditya W Erlangga (G651120714)Jemy Arieswanto (G651120664)Amalia Rahmawati (G651120634)

Bogor, February 16th 2013

Travelling Salesman ProblemChapter 1 & 2

AGENDA

• Introduction• NP-Complete Overview• TSP• Q & A

SECURITY UPDATE

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Find the shortest possible route that visits each city exactly once

and returns to the origin city

TRAVELLING SALESMAN PROBLEM

P, NP, NP-COMPLETE, NP-HARD

Polynomial TimeNondeterministic-Polynomial Time NP-Complete NP-Hard

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

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

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

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

P, NP, NP-COMPLETE, AND NP-HARD CORRELATION

TSP IS NP-HARD

IF P = NP IS SOLVED

U$ 1m

AND CREDITS FROMSCIENTISTS AROUNDTHE WORLD

Millenium PrizeProblem

source: http://www.claymath.org/millennium/P_vs_NP/

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

TSP has never been solved

by scientists and experts so far

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

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

» 25 cities will require:

#cities #tours

5 12

6 60

7 360

8 2,520

9 20,160

10 181,440

310,224,200,866,619,719,680,000

TSP OVERVIEW (3)

8am-10am

2pm-3pm

3am-5am7am-8am10am-1pm

4pm-7pm

Vehicle Routing - Meet customers demands within given time windows using lorries of limited capacity

Depot

6am-9am

6pm-7pm

Much more difficult than TSP

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 foundHowever, there are approximation solutions to solve the TSP:

• Polynomial Time Approximation Scheme (PTAS)• Christofides Algorithm• Double MST Algorithm• Arora’s Algorithm• Mitchell’s Algorithm

QUESTIONS?

THANK YOU