1. Set Based Search Modeling Examples II
Andrew Kuipers
Please include [CPSC433] in the subject line of any s regarding this course.
CPSC 433 Artificial Intelligence

2. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack
0-1 Knapsack Problem
For a given list of n items:
with weights W = <w1, …, wn>
and values V = <v1, …, vn>
we want to maximize the value of a knapsack with capacity C by either placing, or not placing, items from I into C
CPSC 433 Artificial Intelligence

3. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack
Facts
The items in our knapsack, which we can simply represent by their index
F = { 1, …, n }
States
A state is a set of items, where the sum of the weight of the items is less than or equal to capacity C
S = { F’  F | fF’ wf  C }
CPSC 433 Artificial Intelligence

4. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack
Extension Rules
Ext = { A  B | sS  AS | ((s-A)  B)  S }
This basic extension rule definition allows us to perform all sorts of set manipulation, so long as the result is a valid state
ie: adding and removing either single or multiple items from the knapsack, as long as the result weighs less than the maximum capacity C
CPSC 433 Artificial Intelligence

5. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack
Example:
W = { 20, 7, 13, 5 }
V = { 50, 30, 25, 15 }
C = 25
s0 = { }
s1 = { 2, 3 } by A = { }  B = { 2, 3 }
s2 = { 2, 3, 4 } by A = { }  B = { 4 }
s3 = { 1, 4 } by A = { 2, 3 }  B = { 1 }
CPSC 433 Artificial Intelligence

6. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack
Problems with this Model
The control must select between a large number of possible extension rules to apply
Only a single possible solution being manipulated: how can we compare solutions?
How might we define the goal?
While we have correctly modeled the problem, this model does not lend well to a search process
Remember: for a given problem, there are many ways to construct a model
CPSC 433 Artificial Intelligence

7. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Another approach: Genetic Algorithm
Brief Definition: A genetic algorithm (GA) is a search model that mimics the process of natural evolution.
A population of potential solutions (individuals)
Mutate and combine to create new individuals
Use a fitness function to evaluate individuals
CPSC 433 Artificial Intelligence

8. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
GA Operators Brief Introduction
Mutation
Crossover
CPSC 433 Artificial Intelligence

9. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Genetic Algorithm Approach
I = (w1, v1), …, (wn, vn), C = capacity
where w((wi, vi)) = wi and v((wi, vi)) = vi
Facts
F = { { i1, …, im } | 1  j  m ij  I  (j=1..m w(ij))  C }
CPSC 433 Artificial Intelligence

10. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
States
S  2F
* do we need to get any more specific?
Extension Rules
Ext = { A → B | sS  AS | (s-A)BS 
(Mutation(A,B) 
Combination(A,B) }
CPSC 433 Artificial Intelligence

11. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Mutation(A,B) 
A = { P }  B = { P, (P – K)  J }
where K  P , J  I and (K  J) = 
Remove some subset K from the fact P randomly
Replace it with some set of items J that does not contain any items in K
We don’t want to replace the fact, just create a new fact that is a mutation of original fact.
CPSC 433 Artificial Intelligence

12. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Mutation Example:
I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) 
C = 25
A = { (3,4), (7,3), (6,9) } = P
K = { (3,4), (6,9) }  P
J = { (9,7), (11,8) }  I
(K  J) = 
CPSC 433 Artificial Intelligence

13. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Mutation Example:
I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) 
C = 25
A = { (3,4), (7,3), (6,9) } = P
K = { (3,4), (6,9) }  P
J = { (9,7), (11,8) }  I
(K  J) = 
B = { P, (P – K)  J }
= { P, { (7,3), (9,7), (11,8) } }
CPSC 433 Artificial Intelligence

14. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Combination(A,B) 
A = { P, Q }  B = { P, Q, K }
where:
K  (P  Q) 
(K  P)  (K  Q) 
min(|P|,|Q|)  |K|  max(|P|, |Q|)
Use existing facts P and Q to generate a new fact K, which is a combination of P & Q, yet not equal to P or Q, and of size between that of P and Q
CPSC 433 Artificial Intelligence

15. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Combination Example:
I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) 
C = 25
P = { i1, i3 }
Q = { i2, i3, i4 }
P  Q = {i1, i2, i3, i4 }
K = { i2, i3 }  (P  Q)
CPSC 433 Artificial Intelligence

16. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
Combination Example:
I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) 
C = 25
P = { i1, i3 }
Q = { i2, i3, i4 }
P  Q = {i1, i2, i3, i4 }
K = { i2, i3 }  (P  Q)
B = {P, Q, { i2, i3 } }
CPSC 433 Artificial Intelligence

17. CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
How does search proceed in a GA?
What does a search instance look like?
Should a search control only select the best individuals in the state?
Population control, genocide
CPSC 433 Artificial Intelligence