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COSC 4426 Topics in Computer Science II Discrete Optimization

COSC 4426 Topics in Computer Science II Discrete Optimization. Good results with problems that are too big for people or computers to solve completely. http://mathworld.wolfram.com/TravelingSalesmanProblem.html. Difficult problems. hard to represent (what information, what data structures)

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COSC 4426 Topics in Computer Science II Discrete Optimization

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  1. COSC 4426 Topics inComputer Science IIDiscrete Optimization Good results with problems that are too big for people or computers to solve completely http://mathworld.wolfram.com/TravelingSalesmanProblem.html

  2. Difficult problems • hard to represent (what information, what data structures) • no known algorithms • no knownefficientalgorithms • this course: discreet variable problems

  3. Examples • practical examples • scheduling (transportation, timetables,…) • puzzles • crosswords, Sudoku, n Queens • classic examples • SAT: propositional satisfiability problem (independent parameters) • CSP: constraint satisfaction problem (dependent parameters) • TSP: travelling salesman problem (permutations)

  4. SAT: propositional satisfiability problem n propositions, P1, P2, P3, …, Pn What combination of truth values makes a sentence true? Table has 2n rows. n=50, 250 = 1,125,899,906,842,624 n=2; 22 = 4 rows

  5. CSP: constraint satisfaction problem • example – map colouring n countries – 4 possible colours • constraints: adjacent countries different colours • 4n combinations n=13; 413 = 67,108,864 combinations; 25 constraints

  6. TSP: traveling salesman(sic) problem • n cities: what is shortest path visiting all cities, C1, C2, C3, …, Cn once? • (n-1)! routes from home city on complete graph n = 16; (n-1)! = 1,307,674,368,000 C1 n = 5; (n-1)! = 24

  7. Silly Example – one variable • mark in class based on hours attended • number of hours, h, is between 0 and 36 • find optimal attendance (best h) if • mark m is m = 3h - 8 • mark m is m = 20h - h2 • mark m is m = (5h/9 – 10)2 • mark m is m = h3 mod 101 • mark m is m = markarray[h]

  8. m = 3h - 8

  9. m = 3h – 8 m = 20h - h2 m = (5h/9 – 10)2 global optimum

  10. local optimum m = h3 mod 101

  11. m = h3 mod 101 m = markarray[h]

  12. Problem description • fitness function (optimization function, evaluation) – e.g., m = h3 mod 101 • constraints (conditions) – e.g., 0 ≤ h ≤ 36 find global optimum of fitness function without violating constraints OR getting stuck at local optimum • small space: complete search • large space: ?????

  13. Large problems • more possible values • more parameters, n = {n1, n2, n3, …} • more constraints • more complex fitness functions - takes significant time to calculate m = f(n)  too big for exhaustive search

  14. Searchingwithout searching everywhere How to search intelligently/efficiently using information in the problem: -hill climbing -simulated annealing -genetic algorithms -constraint satisfaction -A* - …

  15. Focusing search • assumption – some pattern to the distribution of the fitness function finding the height of land in a forest - can only see ‘local’ structure - easy to find a hilltop but are there other higher hills?

  16. Fitness function distribution • convex – easy – start anywhere, make local decisions

  17. Fitness function distribution • many local maxima make local decisions but don’t get trapped

  18. Course outline • textbook – Michalewicz and Fogel (reasonable price, valuable book) • lectures, notes and ppt presentations • evaluation • assignments • project • tests • final exam

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