1. Lecture 11: Tree Search
© J. Christopher Beck 2008

2. Outline Generate-and-test Partition Relaxation Inference
© J. Christopher Beck 2008

3. Readings
P Ch B.3, B.4
J. Hooker, Integrated Methods for Optimization, 2007
© J. Christopher Beck 2008

4. Beyond Heuristics
What if you want a guarantee of finding the optimal solution?
Or, for a satisfaction problem, that no solution exists?
Imagine a problem with 3 variables
a, b, c є {0, 1}
Simplest thing you can think of?
© J. Christopher Beck 2008

5. Idea #1: Partitioning Easy problem Total Search Space
© J. Christopher Beck 2008
Total Search Space

6. Idea #1: Partitioning
Add constraint to the original problem to form a partition: P1, P2, P3, …
Partitions are easier to solve
Partitions, sub-partitions, sub-sub-partitions
Solution is the best one from all the partitions
© J. Christopher Beck 2008

7. Idea #1: Partitioning Imagine a problem with 3 variables
a, b, c є {0, 1}
minimize
Branch
a = 0
a = 1
b = 0
b = 1
b = 0
b = 1
c = 0
c = 1
c = 0
c = 1
c = 0
c = 1
c = 0
c = 1
100 90
110
115 80 90
100
110
© J. Christopher Beck 2008

8. So …
By now, if you are thinking about partitioning, you should have a question:
(Hint: how many “states” in our generate-and-test example? how many in the tree search?)
© J. Christopher Beck 2008

9. Idea #2: Relaxation
Solve a problem that is easier than the real problem
Relaxation: expand the search space to allow non-solutions to count as solutions
Solution to relaxation is a bound on the real problem
no real solution can be better
© J. Christopher Beck 2008

10. Idea #2: Relaxation
Imagine I have a way to calculate a lower bound on the cost at each node
a = 0
a = 1 50
b = 0
b = 1
b = 0 70 80
Bound
c = 0
c = 1
c = 0 85 95 80
100 90 80
© J. Christopher Beck 2008

11. Branch-and-Bound Partition + Relaxation
Use heuristics to pick a decision to try (“branch”)
partition
Use lower bounds on solutions to “bound” the search
solve a relaxation
© J. Christopher Beck 2008

12. Partitioning + Relaxation
Easy
problem
© J. Christopher Beck 2008
Total Search Space

13. MIP Solving (preview) Relaxation Partition
solve LP, ignoring integrality
gives lower bound (for a minimization problem)
Partition
add linear constraints to force a non-integral integer variable up or down
use relaxation in branching!
© J. Christopher Beck 2008

14. One of the constraints: b < a
Idea #3: Inference
Imagine a problem with 3 variables
a, b, c є {0, 1}
minimize
One of the constraints: b < a
a = 0
a = 1
b = 0
b = 1
b = 0
b = 1
c = 0
c = 1
c = 0
c = 1
c = 0
c = 1
c = 0
c = 1
100 90
110
115 80 90
100
110
© J. Christopher Beck 2008

15. One of the constraints: b < a
Idea #3: Inference
Imagine a problem with 3 variables
a, b, c є {0, 1}
minimize
One of the constraints: b < a
a = 0
b = 0
b = 1
c = 0
c = 1
100 90
110
115
a = 1
You can infer:
a = 0, b = 1
b = 0
c = 0
c = 1
b = 1
100
110
c = 0
c = 1 80 90
© J. Christopher Beck 2008

16. Idea #3: Inference
Based on the constraints in the current partition, derive new constraints that are implied
implied = must be true!
Add new constraints to:
tighten relaxation
reduce partitioning
© J. Christopher Beck 2008

17. Constraint Propagation (preview)
3 variables: v1, v2, v3
D1=D2= {1,3}, D3 = {1,2,3}
all-different(v1,v2,v3)
each variable must be a different value
What can you infer?
© J. Christopher Beck 2008

18. Partition + Relaxation + Inference
Until the partition is easy to solve or has no solution:
infer and add new constraints
calculate relaxation
form sub-partitions
(Tree) Search = Partition + Relaxation + Inference
© J. Christopher Beck 2008

19. A Very Simple Scheduling Problem
Jobs
Processing times
J0R0[15]  J0R1[5] 1
J1R0[10]  J1R1[15]
Draw complete branch-and-bound tree to minimize makespan
Assume that the branches sequence a pair of operations
Try J0Rx → J1Rx first
© J. Christopher Beck 2008

20. Getting Started Take “left” branch first … … Jobs Processing times
J0R0[15]  J0R1[5] 1
J1R0[10]  J1R1[15]
Take “left” branch first
J1R0  J0R0
J0R0  J1R0 ?? … …
© J. Christopher Beck 2008

21. Bounding? Calculate a very simple lower bound at each node
Jobs
Processing times
J0R0[15]  J0R1[5] 1
J1R0[10]  J1R1[15]
Calculate a very simple lower bound at each node
Which nodes will not be visited?
© J. Christopher Beck 2008