SAT problem
► SAT
– Boolean satisfiability problem
F ( x1, x 2, x3, x 4)  x1x 4  x 2 x3
► Find
the assignment to each binary variable so
the expression evaluates to TRUE
► Exhaustive search – EXHAUSTIVE!!!
► Transform the expression to CNF
(Conjunctive Normal Form)
SAT problem
Example
F ( x1, x 2, x3, x 4)  ( x1  x 2) * ( x 2  x 4) * ( x1  x3) * ( x3  x 4)
4 variables – 24=16 combinations to check
F ( x1, x 2,, x100)
100 variables – 2100 ≈ 1030 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
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
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.
TSP
Symmetric
dist(i,j) = dist(j,i)
Assymmetric
dist(i,j) ≠ dist(j,i)
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
TSP
tour can be represented in 2n
different ways (symmetric TSP), and there
are n! ways to permute n numbers
► Every
► |S|=n!/(2n)=(n-1)!/2
► For
n>6 there are more possible solutions to
the TSP than to SAT.
TSP vs. SAT
n
SAT
TSP
3
8
1
4
16
3
5
32
12
6
64
60
7
128
360
8
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!
Problem solving
► In
every problem we must specify:
1. Representation
2. Objective (goal)
3. Evaluation function
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.
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)
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.
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.
Tiling
64 squares = 32*2
62 squares = 31*2
Tiling
64 squares = 32*2
62 squares = 31*2
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!!!
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!
Mr. and Mrs. Smith’s party
Mr. and Mrs. Smith’s party
Mr. and Mrs. Smith’s party
Mr. and Mrs. Smith’s party
Mrs. Smith is
person 4.