1. number partitioning

2. Like subset sum

4. Recently featured in the Crystal Maze!
Given a bag of numbers, can you partition this
into 2 bags such that the sum of the integers
in each bag is equal?
Recently featured in the Crystal Maze!
(Thanks Zoe!)

5. Try it!

6.
Try that!

7. Try that?

8. Try that?

9. Can you think of a 1st test to
carry out to determine if there is no partition?

10. Garey & Johnson “Computers and Intractability”
Given a bag of numbers, can you partition this
into 2 bags such that the sum of the integers
in each bag is equal?
Garey & Johnson “Computers and Intractability”
[SP12] PARTITION
INSTANCE: Finite set A and a size s(a)  Z+ for each a  A
QUESTION: Is there a subset A’  A such that aA’ s(a)  aA-A’ s(a)

12. How complex?

13. Who cares?
Imagine you have 2 machines on the shop floor
You have n activities, of varying durations
Place the activities on the machines to minimise makespan

14. Why just 2-partition?
Why not m-way partitioning?
Is there an optimisation problem?

15. A number of constraint encodings

16. 1st stab
variables
constraints
Questions:
What are the “decision variables”
Can we be more efficient
generate more propagation
Bound or enumerated variables
Is there a better model?
What if it is insoluble?

24. How can I make an IntegerVariable with a domain
that is a set of values, rather than a range?
If I had L[i]  {0,w[i]} and R[i]  {0,w[i]}
could I throw away D[i] and have R[i] ≠ L[i]?
Laura?

28. Look! No D!

29. Value ordering … does it matter
Questions, questions, questions
Decision variables … does it matter?
Heuristics?
Value ordering … does it matter
Bound versus enumerated variables … anyone?
Symmetries
Propagation: what will it do? Will it have an effect?
Size of state space?
How big is the model?
How will search progress?

30. Symmetry
If we are using a static variable ordering heuristic and
If we are using the 0/1 decision variables D[i]
Does it make any difference if we have D[0] = 0 or D[0] = 1
That is, can we half the search space?

31. Can we think of any “side constraints”?
1st stab
Why use CP for numPart?
Can we think of any “side constraints”?
Actually, an important question: justify use of CP

32. We want to minimise the difference between the sum
A 2nd stab
We want to minimise the difference between the sum
of the numbers on the left and the sum of the numbers
on the right
Right!?

33. An optimisation problem Minimise the difference between left and right
A 2nd stab
An optimisation problem
Minimise the difference between left and right

34. Minimize imbalance

35. How do we optimise in CP?
A sequence of decision problems
Branch and bound

37. Demo of Optimize

38. Why is Optimize so slow?

39. When can this propagate?

40. Can we limit the search effort?

42. 3d stab

43. Optimize is a wee bit “clunky”
3d stab
Optimize is a wee bit “clunky”
Can we do better?
Put ½ the numbers on one side

46. 1st stab
variables
constraints

47. Maximize the sum of
Weights on the right
As close to tot/2

48. Demo of OptimizeV2

49. Why was version 2 faster?

50. Could we Optimize without the decision variable D?
I suppose so … could explore this
Replace scalar with sum

51. So?

52. Three stabs!
A fair bit to consider
Has this all been easy?
Is there a better way to do numPart?