COSC 4426 Topics inComputer Science II

Discrete Optimization

Good results with problems that are too bigfor people or computers to solve completely

http://mathworld.wolfram.com/TravelingSalesmanProblem.html

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

P1 P2 P1^P2

F F F

F T T

T F F

T T T

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

CSP: constraint satisfaction problem

example – map colouringn countries – 4 possible colours- constraints: adjacent

countries different colours- 4n combinations

n=13; 413 = 67,108,864 combinations; 25 constraints

TSP: traveling salesman(sic) problem n cities: what is shortest path visiting all

cities, C1, C2, C3, …, Cn once? (n-1)! routes

from home cityon complete graph

n = 16;(n-1)! = 1,307,674,368,000

C1

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 is m = 3h - 82. mark m is m = 20h - h2

3. mark m is m = (5h/9 – 10)2 4. mark m is m = h3 mod 1015. mark m is m = markarray[h]

Marking schemes

-25

0

25

50

75

100

0 6 12 18 24 30 36

attendance

grade

m = 3h - 8

Marking schemes

-25

0

25

50

75

100

0 6 12 18 24 30 36

attendance

grade

m = 3h – 8

m = 20h - h2

m = (5h/9 – 10)2 global optimum

Marking schemes

-25

0

25

50

75

100

0 6 12 18 24 30 36

attendance

grade

m = h3 mod 101 local optimum

Marking schemes

-25

0

25

50

75

100

0 6 12 18 24 30 36

attendance

grade

m = h3 mod 101m = markarray[h]

Marking schemes

-25

0

25

50

75

100

0 6 12 18 24 30 36

attendance

grade

Problem description1. fitness function (optimization function, evaluation) –

e.g., m = h3 mod 1012. constraints (conditions) – e.g., 0 ≤ h ≤ 36find global optimum of fitness function withoutviolating constraints ORgetting stuck at local optimum small space:

complete search1. large space: ?????

Marking schemes

-25

0

25

50

75

100

0 6 12 18 24 30 36

attendance

grade

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

Searchingwithout 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