Travelling Salesman Problem

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  • Travelling Salesman Problem

    1. 1. Travelling Salesman ProblemChapter 1 & 2Raditya W Erlangga (G651120714)Jemy Arieswanto (G651120664)Amalia Rahmawati (G651120634)Bogor, February 16th 2013
    2. 2. AGENDA• Introduction• NP-Complete Overview• TSP•Q&A
    3. 3. TRAVELLING SALESMAN PROBLEMFind the shortest possible routethat visits each city exactly once and returns to the origin city SECURITY UPDATE <ISC SA or IR Number> <Date>
    4. 4. P, NP, NP-COMPLETE, NP-HARD Nondeterministic-Polynomial Time Polynomial Time NP-Complete NP-Hard
    5. 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. 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. 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. 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
    9. 9. P, NP, NP-COMPLETE, AND NP-HARD CORRELATION
    10. 10. TSP IS NP-HARDU$ 1mIF P = NP IS SOLVEDMillenium PrizeProblemAND CREDITS FROMSCIENTISTS AROUNDTHE WORLDsource: http://www.claymath.org/millennium/P_vs_NP/
    11. 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. 12. TSP has never beensolved by scientists and experts so far
    13. 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. 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. 15. TSP OVERVIEW (3) Vehicle Routing - Meet customers demands within given time windows using lorries of limited capacity 10am-1pm 7am-8am 3am-5am4pm-7pm 6pm-7pm Depot 8am-10am 6am-9am 2pm-3pm Much more difficult than TSP
    16. 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
    17. 17. QUESTIONS?
    18. 18. THANK YOU

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