COSC 4426 Topics in Computer Science II Discrete Optimization Good results with problems that are too big for people or computers to solve completely

Difficult problems  hard to represent (what information, what data structures)  no known algorithms  no known efficient algorithms  this course: discreet variable problems

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)

SAT: propositional satisfiability problem P1P1 P2P2 P 1 ^P 2 FFF FTT TFF TTT n propositions, P 1, P 2, P 3, …, P n What combination of truth values makes a sentence true? Table has 2 n rows. n=50, 2 50 = 1,125,899,906,842,624 n=2; 2 2 = 4 rows

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

TSP: traveling salesman (sic) problem  n cities: what is shortest path visiting all cities, C 1, C 2, C 3, …, C n once?  (n-1)! routes from home city on complete graph n = 16; (n-1)! = 1,307,674,368,000 C1C1 n = 5; (n-1)! = 24

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 1.mark m ism = 3h mark m ism = 20h - h 2 3.mark m ism = (5h/9 – 10) 2 4.mark m ism = h 3 mod mark m ism = markarray[h]

m = 3h - 8

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

m = h 3 mod 101 local optimum

m = h 3 mod 101 m = markarray[h]

Problem description 1.fitness function (optimization function, evaluation) – e.g., m = h 3 mod 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: ?????

Large problems  more possible values  more parameters, n = {n 1, n 2, n 3, …}  more constraints  more complex fitness functions - takes significant time to calculate m = f(n)  too big for exhaustive search

Searching without searching everywhere How to search intelligently/efficiently using information in the problem: -hill climbing -simulated annealing -genetic algorithms -constraint satisfaction -A* - …

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?

Fitness function distribution  convex – easy – start anywhere, make local decisions

Fitness function distribution  many local maxima make local decisions but don’t get trapped

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