1. SAT problem SAT – Boolean satisfiability problem
Find the assignment to each binary variable so the expression evaluates to TRUE
Exhaustive search – EXHAUSTIVE!!!
Transform the expression to CNF
(Conjunctive Normal Form)

2. SAT problem Example 4 variables – 24=16 combinations to check
If we check 1 million combinations per second, since the
big bang (10 billion years ago) we would have checked
less than 1% of all combinations

3. SAT problem
Another problem is that after each evaluation we get only 0 (FALSE) or 1 (TRUE), and the result of the evaluation doesn’t tell us what should we do
We don’t know how far we are from the solution

4. TSP TSP – Traveling Salesman Problem
The traveling salesman must visit each town exactly ones and return home covering theminimum distance.
Textbook examples allow connections between all cities, real world problem are different.

5. TSP
Symmetric
dist(i,j) = dist(j,i)
Assymmetric
dist(i,j) ≠ dist(j,i)

6. TSP Size of the search space: permutations of n (=20) cities
Same tours
2 – … – 5 – 6 – 15 – 3 – 11 – 19 – 17
15 – 3 – 11 – 19 – 17 – 2 – … – 6
3 – 11 – 19 – 17 – 2 – … – 6 – 15
Symmetric TSP: reverse tours are the same

7. TSP
Every tour can be represented in 2n different ways (symmetric TSP), and there are n! ways to permute n numbers
|S|=n!/(2n)=(n-1)!/2
For n>6 there are more possible solutions to the TSP than to SAT.

8. TSP vs. SAT n
SAT
TSP 3 8 1 4 16 5 32 12 6 64 60 7
128
360
256
2520 …
n=10
S≈181000
n=20
S ≈1016
n=50
S ≈1062
“Only” 1021 liters
of water on the planet
Better evaluation
for TSP than for SAT!

9. Problem solving In every problem we must specify: 1. Representation
2. Objective (goal)
3. Evaluation function

10. Representation of SAT
n variable that are logical bits, we represent a candidate solution as a binary string of length n.
Each element in the string corresponds to one variable of the problem.
The size of the search space is 2n.

11. Objective
SAT: find the vector of bits such that the Boolean statement is satisfied
TSP: minimize the total distance traveled by the salesman subject to constraint of visiting each city exactly once and returning ti the starting city.
min ∑ dist(x,y)

12. Evaluation function
The evaluation function assigns a number to each candidate solution, indicating its quality.
TSP: map each tour to its corresponding total distance.
When choosing the evaluation function it must have the best value when the solution is found.

13. Evaluation function
SAT: every approximate solution evaluates to FALSE and this doesn’t give us any useful information on how to improve one candidate solution to other, or how to search for better alternative.

14. Tiling
64 squares = 32*2
62 squares = 31*2

15. Tiling
64 squares = 32*2
62 squares = 31*2

16. Matches and triangles
Given 6 matches construct 4 equilateral triangles where the length of each side is equal to the length of a match.
2 triangles, 5 matches (1 match remaining):
We are searching
in the wrong search space!!!

17. Matches and triangles
Given 6 matches construct 4 equilateral triangles where the length of each side is equal to the length of a match.
2 triangles, 5 matches (1 match remaining):
We are searching
in the wrong search space!!!
Solution!

18. Mr. and Mrs. Smith’s party

19. Mr. and Mrs. Smith’s party

20. Mr. and Mrs. Smith’s party

21. Mr. and Mrs. Smith’s party
Mrs. Smith is
person 4.