Modelling

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Modelling

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Title: Modelling

1
Modelling Solving with Constraints

Prof. Toby Walsh
University College Cork/Uppsala University

2
Overview

Introduction to constraint programming
Constraint propagation
Backtracking search
Modelling case studies
Simple recipe
Solving with constraints
Global constraints
Set variables
Branching heuristics

3
Resources

Course links
www.cs.york.ac.uk/tw/Links/csps/
Benchmark problems
www.csplib.org
Constraints solvers
LP based like ECLIPSE, Java based solvers like
JCL,

4
Constraint programming

Dream of declarative programming
State the constraints
Solver finds a solution
Method of choice for many hard combinatorial
problems
Scheduling, assignment, routing,

5
Constraints are everywhere!

No meetings before 10am
Network traffic lt 100 Gbytes/sec
PCB width lt 21cm
Salary gt 45k Euros

6
Constraint satisfaction

Constraint satisfaction problem (CSP) is a triple
ltV,D,Cgt where
V is set of variables
Each X in V has set of values, D_X
Usually assume finite domain
true,false, red,blue,green, 0,10,
C is set of constraints
Goal find assignment of values to variables to
satisfy all the constraints

7
Example CSP

Course timetabling
Variable for each course
CS101, OS102 ..
Domain are possible times
wed9am, fri10am, ..
Constraints
CS101 \ wed9am
Capacity constraints atmost(3,OS102,DB103..,wed
9am)
Lecturer constraints alldifferent(CS101,DB103,
)

8
Constraint optimization

CSP objective function
E.g. objective is Profit Income - Costs
Find assignment of vals to vars that
Satisfies constraints
Maximizes (minimizes) objective
Often solved as sequence of satisfaction problems
Profit gt 0, Profit gt Ans1, Profit gt Ans2,

9
Constraint programming v. Constraint logic
programming

Constraints declaratively specify problem
Logic programming natural approach
Assert constraints, call labelling strategy
(backtracking search predicate)
Imperative functional toolkits
C, Java, CAML,

10
Constraints

Constraints are tuples ltS,Rgt where
S is the scope, X1,X2, Xm
list of variables to which constraint applies
R is relation specifying allowed values (goods)
Subset of D_X1 x D_X2 x x D_Xm
May be specified intensionally or extensionally

11
Constraints

Extensional specification
List of goods (or for tight constraints, nogoods)
Intensional specification
X1 / X2
5X1 6X2 lt X3
alldifferent(X1,X2,X3,X4),

12
Binary v non-binary

Binary constraint
Scope covers 2 variables
E.g. not-equals constraint X1 / X2.
E.g. ordering constraint X1 lt X2
Non-binary constraint
Scope covers 3 or more variables
E.g. alldifferent(X1,X2,X3).
E.g. tour(X1,X2,X3,X4).
Non-binary constraints usually do not
include unary constraints!

13
Constraint graph

Nodes variables
Edge between 2 nodes iff constraint between 2
associated variables
Few constraints, sparse constraint graph
Lots of constraints, dense constraint graph

14
Some non-binary examples

Timetabling
Variables Lecture1, Lecture2,
Values time1, time2,
Constraint that lectures taught by same lecturer
do not conflict
alldifferent(Lecture1,Lecture5,).

15
Some non-binary examples

Scheduling
Variables Job1. Job2,
Values machine1, machine2,
Constraint on number of jobs on each machine
atmost(2,Job1,Job2,,machine1),
atmost(1,Job1,Job2,,machine2).

16
Why use non-binary constraints?

Binary constraints are NP-complete
Any non-binary constraint can be represented
using binary constraints
E.g. alldifferent(X1,X2,X3) is equivalent to X1
/ X2, X1 / X3, X2 / X3
In theory therefore theyre not needed
But in practice, they are!

17
Modelling with non-binary constraints

Benefits include
Compact, declarative specifications
(discussed next)
Efficient constraint propagation
(discussed second)

18
Modelling with non-binary constraints

Consider writing your own alldifferent
constraint
alldifferent().
alldifferent(HeadTail)-
onediff(Head,Tail),
alldifferent(Tail).

onediff(El,).
onediff(El,HeadTail)-
El \ Head,
onediff(El,Tail).

19
Modelling with non-binary constraints

Its possible but its not very pleasant!
Nor is it very compact
alldifferent(X1,Xn) expands into n(n-1)/2
binary not-equals constraints, Xi \ Xj
one non-binary constraint or O(n2) binary
constraints?
And there exist very efficient algorithms
for reasoning efficiently with many specialized
non-binary constraints

20
Constraint solvers

Two main approaches
Systematic, tree search algorithms
Local search or repair based procedures
Other more exotic possibilities
Hybrid algorithms
Quantum algorithms

21
Systematic solvers

Tree search
Assign value to variable
Deduce values that must be removed from
future/unassigned variables
Propagation to ensure some level of consistency
If future variable has no values, backtrack else
repeat
Number of choices
Variable to assign next, value to assign
Some important refinements like nogood learning,
non-chronological backtracking,

22
Local search

Repair based methods
Generate complete assignment
Change value to some variable in a violated
constraint
Number of choices
Violated constraint, variable within it,
Unable to exploit powerful constraint propagation
techniques

23
Constraint propagation

Heart of constraint programming
Most often enforce arc-consistency (AC)
A binary constraint r(X1,X2) is AC iff
for every value for X1, there is a consistent
value (often called support) for X2 and vice
versa
A problem is AC iff every constraint is AC

24
Enforcing arc-consistency

X2 \ X3 is AC
X1 \ X2 is not AC
X21 has no support so can this value can be
pruned
X2 \ X3 is now not AC
No support for X32
This value can also be pruned
Problem is now AC

1
X1
\
1,2
2,3
\
X3
X2
25
Enforcing arc-consistency

Remove all values that are not AC
(i.e. have no support)
May remove support from other values
(often queue based algorithm)
Best AC algorithms (AC7, AC-2000) run in O(ed2)
Optimal if we know nothing else about the
constraints

26
Properties of AC

Unique maximal AC subproblem
Or problem is unsatisfiable
Enforcing AC can process constraints in any order
But order does affect (average-case) efficiency

27
Non-binary constraint propagation

Most popular is generalized arc-consistency (GAC)
A non-binary constraint is GAC iff for every
value for a variable there are consistent values
for all other variables in the constraint
We can again prune values that are not supported
GAC AC on binary constraints

28
GAC on alldifferent

alldifferent(X1,X2,X3)
Constraint is not GAC
X12 cannot be extended
X2 would have to be 3
No value left then for X3
X11 is GAC

2,3
2,3
X2
X3
29
Enforcing GAC

Enforcing GAC is expensive in general
GAC schema is O(dk)
On k-ary constraint on vars with domains of size
d
Trick is to exploit semantics of constraints
Regins all-different algorithm
Achieves GAC in just O(k3/2 d)
On k-ary all different constraint with domains of
size d
Based on finding matching in value graph

30
Other types of constraint propagation

(i,j)-consistency due to Freuder, JACM 85
Non-empty domains
Any consistent instantiation for i variables can
be extended to j others
Describes many different consistency techniques

31
(i,j)-consistency

Generalization of arc-consistency
AC (1,1)-consistency
Path-consistency (2,1)-consistency
Strong path-consistency AC PC
Path inverse consistency (1,2)-consistency

32
Enforcing (i,j)-consistency

problem is (1,1)-consistent (AC)
BUT is not (2,1)-consistent (PC)
X12, X23 cannot be extended to X3
Need to add constraints
not(X12 X23)
not(X12 X33)
Nor is it (1,2)-consistent (PIC)
X12 cannot be extended to X2 X3 (so needs to
be deleted)

1,2
X1
\
\
2,3
2,3
\
X3
X2
33
Other types of constraint propagation

Singleton arc-consistency (SAC)
Problem resulting from instantiating any variable
can be made AC
Restricted path-consistency (RPC)
AC if a value has just one support then any
third variable has a consistent value

34
Comparing local consistencies

Formal definition of tightness introduced by
Debruyne Bessiere IJCAI-97
A-consistency is tighter than B-consistency iff
If a problem is A-consistent -gt it is
B-consistent
We write A gt B

35
Properties

Partial ordering
reflexive A ? A
transitive A ? B B ? C implies A ? C
Defined relations
tighter A gt B iff A ? B not B
? A
incomparable A _at_ B iff neither A ? B
nor B ? A

36
Comparison of consistency techniques

Exercise for the reader, prove the following
identities!
Strong PC gt SAC gt RPC gt AC
NB gaps can reduce search exponentially!

37
Which to choose?

For binary constraints, AC is often chosen
Space efficient
Just prune domains (cf PC)
Time efficient
For non-binary constraints GAC is often chosen
If we can exploit the constraint semantics to
keep it cheap!

38
Why consider these other consistencies?

Promising experimental results
Useful pruning for their additional cost
Theoretical value
E.g. GAC on non-binary constraints may exceed SAC
on equivalent binary model

39
Maintaining a local consistency property

Tree search
Assign value to variable
Enforce some level of local consistency
Remove values/add new constraints
If any future variable has no values, backtrack
else repeat
Two popular algorithms
Maintaining arc-consistency (MAC)
Forward checking (only enforce AC on instantiated
variable)

40
Modelling case study all interval series

Results due to Simonis, Puget Regin

41
All interval series

Prob007 at www.csplib.org
Comes from musical composition
Traced back to Alban Berg
Extensively used by Ernst Krenek
Op.170 Quaestio temporis

42
All interval series

Take the 12 standard pitch classes
c, c, d, ..
Represent them by numbers 0, .., 11
Find a sequence so each occurs once
Each difference occurs once

43
All interval series

Can generalize to any n (not just 12)
Find Sn, a permutation of 0,n)
such that Sn1-Sn are all distinct
Finding one solution is easy

44
All interval series

Can generalize to any n (not just 12)
Find Sn, a permutation of 0,n) such that
Sn1-Sn are all distinct
Finding one solution is easy
n,1,n-1,2,n-2,.., floor(n/2)2,floor(n/2)-1,flo
or(n/2)1,floor(n/2)
Giving the differences n-1,n-2,..,2,1
Challenge is to find all solutions!

45
Basic recipe

Devise basic CSP model
What are the variables? What are the constraints?
Introduce auxiliary variables if needed
Consider dual or combined models
Break symmetry
Introduce implied constraints

46
Basic CSP model

What are the variables?

47
Basic CSP model

What are the variables?
Si j if the ith note is j
What are the constraints?

48
Basic CSP model

What are the variables?
Si j if the ith note is j
What are the constraints?
Si in 0,n)
All-different(S1,S2, Sn)
Forall ilti Si1 - Si / Si1 - Si

49
Basic recipe

Devise basic CSP model
What are the variables? What are the constraints?
Introduce auxiliary variables if needed
Consider dual or combined models
Break symmetry
Introduce implied constraints

50
Improving basic model

Introduce auxiliary variables?
Are there any loose or messy constraints we could
better (more compactly?) express via some
auxiliary variables?

51
Improving basic model

Introduce auxiliary variables?
Yes, variables for the pairwise differences
Di Si1 - Si
Now post single large all-different constraint
Di in 1,n-1
All-different(D1,D2,Dn-1)

52
Basic recipe

Devise basic CSP model
What are the variables? What are the constraints?
Introduce auxiliary variables if needed
Consider dual or combined models
Break symmetry
Introduce implied constraints

53
Break symmetry

Does the problem have any symmetry?

54
Break symmetry

Does the problem have any symmetry?
Yes, we can reverse any sequence
S1, S2, Sn is an all-inverse series
Sn, , S2, S1 is also
How do we eliminate this symmetry?

55
Break symmetry

Does the problem have any symmetry?
Yes, we can reverse any sequence
S1, S2, , Sn is an all-inverse series
Sn, , S2, S1 is also
How do we eliminate this symmetry?
Order first and last difference
D1 lt Dn-1

56
Break symmetry

Does the problem have any other symmetry?

57
Break symmetry

Does the problem have any other symmetry?
Yes, we can invert the numbers in any sequence
0, n-1, 1, n-2, map x onto n-1-x
n-1, 0, n-2, 1,
How do we eliminate this symmetry?

58
Break symmetry

Does the problem have any other symmetry?
Yes, we can invert the numbers in any sequence
0, n-1, 1, n-2, map x onto n-1-x
n-1, 0, n-2, 1,
How do we eliminate this symmetry?
S1 lt S2

59
Performance

Basic model is poor
Improved model able to compute all solutions up
to n14 or so
GAC on all-different constraints very beneficial
As is enforcing GAC on Di Si1-Si
This becomes too expensive for large n
So use just bounds consistency (BC) for larger n

60
Modelling case study Langfords problem

Model due to Barbara Smith

61
Outline

Introduction
Langfords problem
Modelling it as a CSP
Basic model
Refined model
Experimental Results
Conclusions

62
Recipe

Create a basic model
Decide on the variables
Introduce auxiliary variables
For messy/loose constraints
Consider dual, combined or 0/1 models
Break symmetry
Add implied constraints
Customize solver
Variable, value ordering

63
Langfords problem

Prob024 _at_ www.csplib.org
Find a sequence of 8 numbers
Each number 1,4 occurs twice
Two occurrences of i are i numbers apart
Unique solution
41312432

64
Langfords problem

L(k,n) problem
To find a sequence of kn numbers 1,n
Each of the k successive occrrences of i are i
apart
We just saw L(2,4)
Due to the mathematician Dudley Langford
Watched his son build a tower which solved L(2,3)

65
Langfords problem

L(2,3) and L(2,4) have unique solutions
L(2,4n) and L(2,4n-1) have solutions
L(2,4n-2) and L(2,4n-3) do not
Computing all solutions of L(2,19) took 2.5
years!
L(3,n)
No solutions 0ltnlt8, 10ltnlt17, 20, ..
Solutions 9,10,17,18,19, ..
A014552
Sequence 0,0,1,1,0,0,26,150,0,0,17792,108144,0,
0,39809640,326721800,
0,0,256814891280,2636337861200

66
Basic model

What are the variables?

67
Basic model

What are the variables?
Variable for each occurrence of a number
X11 is 1st occurrence of 1
X21 is 1st occurrence of 2
..
X12 is 2nd occurrence of 1
X22 is 2nd occurrence of 2
..
Value is position in the sequence

68
Basic model

What are the constraints?
Xij in 1,nk
Xij1 iXij
Alldifferent(X11,..Xn1,X12,..Xn2,..,X1k,..Xnk)

69
Recipe

Create a basic model
Decide on the variables
Introduce auxiliary variables
For messy/loose constraints
Consider dual, combined or 0/1 models
Break symmetry
Add implied constraints
Customize solver
Variable, value ordering

70
Break symmetry

Does the problem have any symmetry?

71
Break symmetry

Does the problem have any symmetry?
Of course, we can invert any sequence!

72
Break symmetry

How do we break this symmetry?

73
Break symmetry

How do we break this symmetry?
Many possible ways
For example, for L(3,9)
Either X92 lt 14 (2nd occurrence of 9 is in 1st
half)
Or X9214 and X82lt14 (2nd occurrence of 8 is in
1st half)

74
Recipe

Create a basic model
Decide on the variables
Introduce auxiliary variables
For messy/loose constraints
Consider dual, combined or 0/1 models
Break symmetry
Add implied constraints
Customize solver
Variable, value ordering

75
What about dual model?

Can we take a dual view?

76
What about dual model?

Can we take a dual view?
Of course we can, its a permutation!

77
Dual model

What are the variables?
Variable for each position i
What are the values?

78
Dual model

What are the variables?
Variable for each position i
What are the values?
If use the number at that position, we cannot use
an all-different constraint
Each number occurs not once but k times

79
Dual model

What are the variables?
Variable for each position i
What are the values?
Solution 1 use values from 1,nk with the
value inj standing for the ith occurrence of j
Now want to find a permutation of these numbers
subject to the distance constraint

80
Dual model

What are the variables?
Variable for each position i
What are the values?
Solution 2 use as values the numbers 1,n
Each number occurs exactly k times
Fortunately, there is a generalization of
all-different called the global cardinality
constraint (gcc) for this

81
Global cardinality constraint

Gcc(X1,..Xn,l,u) enforces values used by Xi to
occur between l and u times
All-different(X1,..Xn) Gcc(X1,..Xn,1,1)
Regins algorithm enforces GAC on Gcc in O(n2.d)

82
Dual model

What are the constraints?
Gcc(D1,Dkn,k,k)
Distance constraints?

83
Dual model

What are the constraints?
Gcc(D1,Dkn,k,k)
Distance constraints
Dij then Dij1j

84
Combined model

Primal and dual variables
Channelling to link them
What do the channelling constraints look like?

85
Combined model

Primal and dual variables
Channelling to link them
Xijk implies Dki

86
Solving choices?

Which variables to assign?
Xij or Di

87
Solving choices?

Which variables to assign?
Xij or Di, doesnt seem to matter
Which variable ordering heuristic?
Fail First or Lex?

88
Solving choices?

Which variables to assign?
Xij or Di, doesnt seem to matter
Which variable ordering heuristic?
Fail First very marginally better than Lex
How to deal with the permutation constraint?
GAC on the all-different
AC on the channelling
AC on the decomposition

89
Solving choices?

Which variables to assign?
Xij or Di, doesnt seem to matter
Which variable ordering heuristic?
Fail First very marginally better than Lex
How to deal with the permutation constraint?
AC on the channelling is often best for time

90
Global constraints
91
Non-exhaustive catalog

Order constraints
Constraints on values
Partitioning constraints
Timetabling constraints
Graph constraints
Scheduling constraints
Bin-packing constraints

92
Global constraints

It isnt just all-different!
Many constraints specialized to application
domains
Scheduling
Packing
..

93
Order constraints

min(X,Y1,..,Yn) and max(X,Y1,..Yn)
X lt minimum(Y1,..,Yn)
X gt maximum(Y1,..Yn)

94
Order constraints

min_n(X,n,Y1,..Ym) and max_n(X,n,Y1,..,Ym)
X is nth smallest value in Y1,..Ym
X is nth largest value in Y1,..Ym

95
Value constraints

among(N,Y1,..,Yn,val1,..,valm)
N vars in Y1,..,Yn take values val1,..valm
e.g. among(2,1,2,1,3,1,5,3,4,5)

96
Value constraints

among(N,Y1,..,Yn,val1,..,valm)
N vars in Y1,..,Yn take values val1,..valm
e.g. among(2,1,2,1,3,1,5,3,4,5)
count(n,Y1,..,Ym,op,X) where op is ,lt,gt,/,lt
or gt
relation Yi op X holds n times
among(n,Y1,..,Ym,k)
count(n,Y1,..,Ym,,k)

97
Value constraints

balance(N,Y1,..,Yn)
N occurrence of more frequent value -
occurrence of least frequent value
E.g balance(2,1,1,1,3,4,2)

98
Value constraints

balance(N,Y1,..,Yn)
N occurrence of more frequent value -
occurrence of least frequent value
E.g balance(2,1,1,1,3,4,2)
all-different(Y1,..,Yn) gt balance(0,Y1,..,Yn
)

99
Value constraints

min_nvalue(N,Y1,..,Yn) and max_nvalue(N,Y1,..,Y
n)
least (most) common value in Y1,..,Yn occurs N
times
E.g. min_nvalue(2,1,1,2,2,2,3,3,5,5)
Can replace multiple count or among constraints

100
Value constraints

common(X,Y,X1,..,Xn,Y1,..,Ym)
X vars in Xi take a value in Yi
Y vars in Yi take a value in Xi
E.g. common(3,4,1,9,1,5,2,1,9,9,6,9)

101
Value constraints

common(X,Y,X1,..,Xn,Y1,..,Ym)
X vars in Xi take a value in Yi
Y vars in Yi take a value in Xi
E.g. common(3,4,1,9,1,5,2,1,9,9,6,9)
among(X,Y1,..,Yn,val1,..,valm)
common(X,Y,X1,..,Yn,val1,..,valm)

102
Value constraints

same(X1,..,Xn,Y1,..,Yn)
Yi is a permutation of Xi

103
Value constraints

same(X1,..,Xn,Y1,..,Yn)
Yi is a permutation of Xi
used_by(X1,..,Xn,Y1,..,Ym)
all values in Yi are used by vars in Xi
mgtm

104
Value constraints

same(X1,..,Xn,Y1,..,Yn)
Yi is a permutation of Xi
used_by(X1,..,Xn,Y1,..,Ym)
all values in Yi are used by vars in Xi
mgtm
on n values
alldifferent(X1,..,Xn)same(X1,..,Xn,1,..,n
)
used_by(X1,..,Xn,1,..,n)

105
Partitioning constraints

all-different(X1,..,Xn)

106
Partitioning constraints

all-different(X1,..,Xn)
Other flavours
all-different_except_0(X1,..,Xn)
Xi/Xj unless XiXj0
0 is often used for modelling purposes as dummy
value
Dont use this slab
Dont open this bin ..

107
Partitioning constraints

all-different(X1,..,Xn)
Other flavours
symmetric-all-different(X1,..,Xn)
Xi/Xj and Xij iff Xji
Very common in practice
Team i plays j iff Team j plays i..

108
Partitioning constraints

nvalue(N,X1,..,Xn)
Xi takes N different values
all-different(X1,..,Xn) nvalue(n,X1,..,Xn)

109
Partitioning constraints

nvalue(N,X1,..,Xn)
Xi takes N different values
all-different(X1,..,Xn) nvalue(n,X1,..,Xn)
gcc(X1,..,Xn,Lo,Hi)
values in Xi occur between Lo and Hi times
all-different(X1,..,Xn)gcc(X1,..,Xn,1,1)

110
Timetabling constraints

change(N,X1,..,Xn),op) where op is
,lt,gt,lt,gt,/
Xi op Xi1 holds N times
E.g. change(3,4,4,3,4,1,/)
You may wish to limit the number of changes of
classroom, shifts,

111
Timetabling constraints

longest_changes(N,X1,..,Xn),op) where op is
,lt,gt,lt,gt,/
longest sequence Xi op Xi1 is of length N
E.g. longest_changes(2,4,4,4,3,3,2,4,1,1,1,)
You may wish to limit the length of a shift
without break,

112
Graph constraints

Tours in graph a often represented by the
successors
X1,..,Xn means from node i we go to node Xi

113
Graph constraints

Tours in graph a often represented by the
successors
X1,..,Xn means from node i we go to node Xi
E.g. 2,1,5,3,4 represents the 2 cycles
(1)-gt(2)-gt(1) and (3)-gt(5)-gt(4)-gt(3)

114
Graph constraints

cycle(N,X1,..,Xn)
there are N cycles in Xi
e.g. cycle(2,2,1,5,3,4) as we have the 2
cycles
(1)-gt(2)-gt(1) and (3)-gt(5)-gt(4)-gt(3)
Useful for TSP like problems (e.g. sending
engineers out to repair phones)

115
Scheduling constraints

cummulative(S1,..,Sn,D1,..,Dn,E1,..,En,H1,.
.,Hn,L)
schedules n (concurrent) jobs, each with a height
Hi
ith job starts at Si, runs for Di and ends at Ei
EiSiDi
at any time, accumulated height of running jobs
is less than L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25
116
Scheduling constraints

coloured_cummulative(S1,..,Sn,D1,..,Dn,E1,..,
En,C1,..,Cn,L)
schedules n (concurrent) each with a colour Ci
no more than L colours running at any one time

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25
117
Scheduling constraints

cycle_cummulative(m,S1,..,Sn,D1,..,Dn,E1,..,E
n,H1,..,Hn,L)
schedules n (concurrent) jobs, each with a height
Hi onto a cyclic schedule of length m

118
Scheduling constraints

cummulatives(M1,,,Mn,S1,..,Sn,D1,..,Dn,E1,.
.,En,H1,..,Hn,L1,..,Lm)
schedules n (concurrent) jobs, each with a height
Hi onto one of m machines
ith runs on Mi
accumulated height of running jobs on machine i
lt Li

119
Scheduling constraints

cummulatives(M1,..Mn,S1,..,Sn,D1,..,Dn,E1,.
.,En,H1,..,Hn,L1,..,Lm)

Machine 1
Machine 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25
120
Scheduling constraints

coloured_cummulatives(M1,,,Mn,S1,..,Sn,D1,..,
Dn,E1,..,En,C1,..,Cn,L1,..,Lm)
schedules n (concurrent) jobs, each with a colour
i onto one of m machines
ith runs on Mi
number of colours of running jobs on machine i
lt Li

121
Bin-packing constraints

bin_packing(capacity,B1,..,Bn,w1,..,wn)
for each bin j, sum_Bij wi lt capacity

122
Bin-packing constraints

bin_packing(capacity,B1,..,Bn,w1,..,wn)
for each bin j, sum_Bij wi lt capacity
special case of cummulative with task durations1

123
Misc constraints

element(Index,a1,..,an,Var)
Vara_Index
constraint programmings answer to arrays!
e.g. element(Item,10,23,12,15,Cost)

124
Modelling set variables
125
Motivation

Sets are useful when
We dont know how many objects we will have
e.g. set of items go in a bin
We have symmetrical objects
e.g. items are symmetric and we dont want to
consider all their permutations
And in many other situations!

126
Outline

Representing set variables
Bounds
Characteristic functions
Constraining set variables
Primitive constraints
Global constraints

127
Set variables

Representing sets
Domain of values powerset
Exponential space would be needed to represent
this extensional

128
Set variables

Representing sets
Domain of values powerset
Exponential space would be needed to represent
this extensional
Compromise just represent upper and lower bound
E.g. subseteq X subseteq 1,2
X in ,1,2,1,2

129
Set variables

Representing sets
Domain of values powerset
Exponential space would be needed to represent
this extensional
Compromise just represent upper and lower bound
E.g. subseteq X subseteq 1,2
X in ,1,2,1,2
Tradeoff
Cannot represent disjunction
E.g. X is 1 or 2 but not 1,2

130
Alternative representation

Characteristic function
i in X iff Xi1

131
Alternative representation

Characteristic function
i in X iff Xi1
E.g. 1 subseteq X subseteq 1,2,3

132
Alternative representation

Characteristic function
i in X iff Xi1
E.g. 1 subseteq X subseteq 1,2,3
X11
X2 in 0,1, X3 in 0,1
X4X5..0

133
Primitive constraints

X subset Y
X subseteq Y
a in X
X Y intersect Z
X Y union Z
X Y - Z
X

134
Bounds consistency

Analogous to bounds consistency on ordered finite
domains
Given constraint, C over X1,..,Xn
C is BC if for each Xi,
a in glb(Xi) iff a is in some solution
a in lub(Xi) iff a is in all solutions

135
Bounds consistency

Lub(A union B) gt Lub(A) union Lub(B)
Glb(A union B) gt Glb(A) union Glb(B)

136
Bounds consistency

Lub(A union B) gt Lub(A) union Lub(B)
Glb(A union B) gt Glb(A) union Glb(B)
This last rule is a safe approximation
Glb(A union B) superseteq Glb(A) union Glb(B)

137
Bounds consistency

Lub(A union B) gt Lub(A) union Lub(B)
Glb(A union B) gt Glb(A) union Glb(B)
Lub(A intersect B) gt Lub(A) intersect Lub(B)
Glb(A intersect B) gt Glb(A) intersect Glb(B)

138
Bounds consistency

Lub(A union B) gt Lub(A) union Lub(B)
Glb(A union B) gt Glb(A) union Glb(B)
Lub(A intersect B) gt Lub(A) intersect Lub(B)
Glb(A intersect B) gt Glb(A) intersect Glb(B)
The third rule is a safe approximation
Lub(A intersect B) subseteq
Lub(A)
intersect Lub(B)

139
Bounds consistency

Lub(A union B) gt Lub(A) union Lub(B)
Glb(A union B) gt Glb(A) union Glb(B)
Lub(A intersect B) gt Lub(A) intersect Lub(B)
Glb(A intersect B) gt Glb(A) intersect Glb(B)
A subseteq B gt Lub(A) subseteq Lub(B),
Glb(A) subseteq
Glb(B)

140
Bounds consistency

Lub(A union B) gt Lub(A) union Lub(B)
Glb(A union B) gt Glb(A) union Glb(B)
Lub(A intersect B) gt Lub(A) intersect Lub(B)
Glb(A intersect B) gt Glb(A) intersect Glb(B)
A subseteq B gt Lub(A) subseteq Lub(B),
Glb(A) subseteq
Glb(B)
A B gt A subseteq B, B subseteq A

141
Bounds consistency

Apply these rules exhaustively
This computes safe approximations
Lubs are correct or too large
Glbs are correct or too small
If a set of constraints have a solution
Exists unique lub and glb for the set variables
That satisfy these rules

142
Intervals v Characteristic functions

It doesnt really matter!
Theorem
A set of constraints in normal form is BC iff the
characteristic function representation is AC

143
Intervals v Characteristic functions

It doesnt really matter!
Theorem
A set of constraints in normal form is BC iff the
characteristic function representation is AC
NB characteristic function is 0/1 model so AC is
the same as bounds consistency!

144
Global constraints