1. The Theory and Practice of Constraint Programming: An Overview
Brahim Hnich
Faculty of Computer Science
Izmir University of Economics
Izmir, Turkey

2. Quotation
“Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it.”
Eugene C. Freuder, Constraints, April 1997
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3. Caveat In this talk: Constraint programming for combinatorial problems
“Programming” refers to its roots in computer science (programming languages)
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4. Outline Modelling Constraint propagation Search
Demo: Lot-sizing with stochastic non-stationary demand
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5. A Puzzle Place the numbers 1 through 8 in the nodes such that: ?
Each number appears exactly once
No connected nodes have consecutive numbers ?
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6. Modeling Each node  a decision variable
{1, …, 8}  values in the domain of each variable
No consecutive numbers  a constraint
(vi, vj)  |vi – vj| > 1
All values used  a clique of not-equals constraints
forall i<j. vi ≠ vj,
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7. Modeling Each node  a decision variable
{1, …, 8}  values in the domain of each variable
No consecutive numbers  a constraint
(vi, vj)  |vi – vj| > 1
All values used 
forall i<j. vi ≠ vj,
Or more compactly, all-different[v1,…,v8]
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8. Heuristic Search
{1, 2, 3, 4, 5, 6, 7, 8} ? 1 8
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9. Inference/Propagation
{1, 2, 3, 4, 5, 6, 7, 8} ? 1 8
{1, 2, 3, 4, 5, 6, 7, 8}
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10. Inference/Propagation
{3, 4, 5, 6}
{3, 4, 5, 6} ? 3 6
{3, 4, 5, 6, 7}
{2, 3, 4, 5, 6} 7 1 8 2 4 5
{3, 4, 5, 6}
{3, 4, 5, 6}
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11. An OPL Model int N=8; struct edge{int x; int y;};
{edge} Edges ={<1,2>, <1,3>,…,<7,8>};
range Nodes 1..N;
range Values 1..N;
var int Solution[Nodes] in Values;
solve{
forall(e in Edges)
abs(Solution[e.x] - Solution[e.y]) >1;
alldifferent(Solution); };
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12. The Core of Constraint Computation
Modelling
Solving
Heuristic
Search
Propagation
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13. ANY QUESTIONS?

14. Modeling

15. Finite-domain Variables
To each variable x is associated a set of values called its domain
e.g x Є {1,3,11}
Each variable must take a value from its domain
That domain is updated as decisions are made
A domain may only shrink (no value is ever added)
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16. Constraint Satisfaction Problems
CSP: (X, D, C)
X = {x1, x2,…, xn} variables
D = {d1, d2,…,dn} domains (finite)
C = {c1,c2,…,ce } constraints
c ЄC var(c) = {xi, xj,…, xk} scope
rel(c) ⊆ di x dj x .. x dk permitted tuples
Solution:
assignment satisfying every constraint
NP-complete task
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17. CSP: Relevance Many problems can be represented as CSP:
Real-life applications
Production planning
Staff scheduling
Resource allocation
Circuit design
Option trading
DNA sequencing
...
Artificial Intelligence
temporal reasoning
Control Theory
controllers for sensory based robots
Concurency
process comm. and synchr.
Computer Graphics
geometric coherence
Database Systems
constraint databases
Bioinformatics
sequence alignment
Operations research
optimization
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18. Constraint Programming
CP:
provides a platform for solving CSPs
proven useful in many real applications
Platform:
set of common structures to reuse
best known algorithms for propagation & solving
Two stages:
modelling
solving
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19. Language of Constraints
The usual relational operators (<, =, >, ≤, …)
… including ≠
Linear and nonlinear constraints
Logical connectives (→, ↔, ┐, …)
Set constraints (subset, union, intersection,…)
“Global constraints”: constraints capturing a common substructure (pattern) of combinatorial problems
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20. Map Colouring Variables: F, N, S Values: { }
Constraints: N ≠ S ≠ F ≠ N N S F
A solution: F N S
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21. Word Design Problem
This problem has its roots in Bioinformatics and Coding Theory.
Problem: find as large a set S of strings (words) of length 8 over the alphabet W = { A,C,G,T } with the following properties:
Each word in S has 4 symbols from { C,G };
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22. Word Design Problem
Problem: find as large a set S of strings (words) of length 8 over the alphabet W = { A,C,G,T } with the following properties:
Each word in S has 4 symbols from { C,G };
Each pair of distinct words in S differ in at least 4 positions; and
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23. Word Design Problem Each word in S has 4 symbols from { C,G };
Problem: find as large a set S of strings (words) of length 8 over the alphabet W = { A,C,G,T } with the following properties:
Each word in S has 4 symbols from { C,G };
Each pair of distinct words in S differ in at least 4 positions; and
Each pair of words x and y in S (where x and y may be identical) are such that xR and yC differ in at least 4 positions.
(x1,…,x8)R = ( x8,…,x1 ) is the reverse of ( x1,…,x8 )
(y1,…,y8)C is the Watson-Crick complement of ( y1,…,y8 ), i.e. the word where each A is replaced by a T and vice versa and each C is replaced by a G and vice versa.
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24. A Solution S= { AAGCCGTT, TACGCGAT}
Each word in S has 4 symbols from { C,G };
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25. A Solution S= { AAGCCGTT, TACGCGAT}
Each pair of distinct words in S differ
in at least 4 positions
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26. A Solution S= { AAGCCGTT, TACGCGAT} SR= { TTGCCGAA, TAGCGCAT}
SC= { TTCGGCAA, ATGCGCTA}
Each pair of words x and y in S
(where x and y may be identical) are such that
xR and yC differ in at least 4 positions
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27. A Matrix Model M 1 2 3 4 5 6 7 8 … m Each row represents a word in S
{ A,C,G,T } … m
Each row represents a word in S
M[i,j]: a decision variable with domain { A,C,G,T }
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28. A Matrix Model
Expressing that each word in S has 4 symbols from { C,G } M 1 2 3 4 5 6 7 8
{ A,C,G,T } … m
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29. A Matrix Model M 1 2 3 4 5 6 7 8 … m For each row r
{ A,C,G,T } … m
For each row r
sum (p in 1..8) //channelling constraints
(M[r,p]=C or M[r,p]=G) = 4
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30. A Matrix Model …
Each pair of distinct words in S differ in at least 4 positions M 1 2 3 4 5 6 7 8
{ A,C,G,T } … m
For each distinct rows r1 and r2
sum(p in 1..8) (M[r1,p] ≠ M[r2,p]) >= 4
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31. A Matrix Model …
xR and yC differ in at least 4 positions. MC 1 2 3 4 5 6 7 8
{ A,C,G,T } … m
Introduce a “compliment matrix”
Each pair //channelling constraints
<M[i,j], MC[i,j]> in {<C,G>, <G,C>, <A,T>, <T,A>}
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32. A Matrix Model …
xR and yC differ in at least 4 positions. MC 1 2 3 4 5 6 7 8
{ A,C,G,T } … m
For each rows r1 and r2
sum(p in 1..8) (M[r1,9-p] ≠ MC[r2,p]) >= 4
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33. ANY QUESTIONS?

34. Constraint Propagation

35. Constraint Propagation
General principle
Consistency
Filtering on simple constraints
Filtering on global constraints
Conclusion
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36. General Principle
Each type of constraint relies on its own specific filtering algorithm to filter out (locally) inconsistent variable assignment
The communication between the different filtering algorithms takes place through the domains of the variables
It makes it easy to have different types of constraints work together
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37. General Principle
{1, 2, 3, 4, 5, 6, 7, 8}
Constraint network ? ≠ ≠ ≠ ≠ ≠ ≠ ≠ 1 8 ≠ ≠ ≠ ≠ ≠ ≠ ≠
{1, 2, 3, 4, 5, 6, 7, 8} ≠ ≠ ≠
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38. General Principle This propagation necessarily terminates
The results is the same regardless of the order in which we consider the constraints
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39. Triggering the constraints
A constraint is woken up by one of its variables.
domain event: anytime the domain changes
(i.e. some value has been removed)
range event: only when the minimum or maximum value in the domain has changed
(e.g. for x ≤ y constraint)
value event: only when the domain is reduced to a single value
(e.g. for x≠y constraint)
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40. Constraint Propagation
General principle
Consistency
Filtering on simple constraints
Filtering on global constraints
Conclusion
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41. Consistency We reason locally about a constraint, removing
inconsistent values from domains
We can often characterize the level of consistency achieved by a filtering algorithm
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42. Generalized Arc Consistency
Given a constraint C on the variables X
A support for Xi = vj on C is a partial assignment containing Xi = vj that satisfies C.
A variable Xi is generalized arc consistent (GAC) on C iff every value in D(Xi) has support on C.
C is GAC iff each constrained variable is GAC on C.
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43. Bounds Consistency Given a constraint C on the variables X
A bound support on C is a support where the interval [min(Xi); max(Xi)] is substituted for the domain of each constrained variable Xi.
A variable Xi is bound consistent (BC) on C if min(Xi) and max(Xi) have bound support on C.
C is BC iff all constrained variables are BC on C.
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44. Consistency: BC example
Consider 4x + 3y - 2z = 10
with Dx = Dy = Dz = {0, 1, , 9}
Maintaining BC reduces domains to
Dx = {0, 1, , 7},
Dy = Dz = {0, 1, , 9}
BC is triggered on range event
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45. Consistency: GAC example
Consider 4x + 3y - 2z = 10
with Dx = Dy = Dz = {0, 1, , 9}
Maintaining GAC reduces domains to
Dx = {0, 1, , 7},
Dy = {0, 2, 4, 6, 8},
Dz = {0, 1, , 9}
GAC is triggered on domain event
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46. Constraint Propagation
General principle
Consistency
Filtering on simple constraints
Filtering on global constraints
Conclusion
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47. Disequalities: Filtering Algorithm
Consider x≠y with Dx = Dy = {0, 1, , 9}
Only once one of the variables is fixed may we remove any value from the domain of the other:
Dx = {3} → Dy = {0, , 2, 4, , 9}
achieves GAC
triggered on value event
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48. Inequalities: Filtering Algorithm
Consider x<y with Dx = Dy = {0, 1, 2, 3}
Only once one of the variables’ bound is changed may we remove any value from the domain of the other:
Dx = {0, 1, 2}
Dy = {1, 2, 3}
achieves BC (which is equivalent to GAC because of monotonicity)
triggered on range event
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49. Constraint Propagation
General principle
Consistency
Filtering on simple constraints
Filtering on global constraints
Conclusion
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50. Example: all-different
Var: F, N, S; Val: { }; Ctrs: N ≠ S ≠ F ≠ N
F { }
S { }
N { } ≠
3 binary constraints, they are GAC, no pruning
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51. Example: all-different
Var: F, N, S; Val: { }; Ctrs: N ≠ S ≠ F ≠ N
F { }
S { }
N { } ≠
Using binary disequalities, this inconsistency goes undetected
3 binary constraints, they are GAC, no pruning
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52. Example: all-different
Var: F, N, S; Val: { }; Ctrs: N ≠ S ≠ F ≠ N
F { }
S { }
N { } ≠
We can do something simple:
Count the number of variables, n
Count the number of values in the union of their domains, m
If n > m then no solution can possibly be found
3 binary constraints, they are GAC, no pruning
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53. Example: all-different
Var: F, N, S; Val: { }; Ctrs: N ≠ S ≠ F ≠ N
F { }
S { }
N { } ≠
We can do something simple:
Count the number of variables, n
Count the number of values in the union of their domains, m
If n > m then no solution can possibly be found
...still fooled by
Dx = Dy = Dz = {a, b}, Dw = {c, d}
3 binary constraints, they are GAC, no pruning
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54. Example: all-different
Var: F, N, S; Val: { }; Ctrs: N ≠ S ≠ F ≠ N
F { }
S { }
N { } ≠
all-different
F { }
S { }
N { }
logically equivalent
3 binary constraints, they are GAC, no pruning
1 ternary constraint, not GAC, GAC pruning ® empty domain no solution!!
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55. All-different: A Filtering Algorithm
Build the corresponding bipartite graph x1 x2 x3 x4 x5 x6 1 2 3 4 5 6 7
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56. All-different: A Filtering Algorithm
Build the corresponding bipartite graph
∃ solution iff ∃ matching covering all the variables x1 x2 x3 x4 x5 x6 1 2 3 4 5 6 7
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57. All-different: A Filtering Algorithm
Build the corresponding bipartite graph
∃ solution iff ∃ matching covering all the variables
filtering
find alternating cycles and paths
remove inconsistent values (useless edges) [X4=2]
fix variables (vital edges) [X4=4] x1 x2 x3 x4 x5 x6 1 2 3 4 5 6 7
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58. All-different: A Filtering Algorithm
Build the corresponding bipartite graph
∃ solution iff ∃ matching covering all the variables
starting from the current matching at each call makes the algorithm incremental
achieves GAC x1 x2 x3 x4 x5 x6 1 2 3 4 5 6 7
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59. Cardinality: A Filtering Algorithm
distribute([c1, , cm],[v1, , vm],[x1, ,xn]),
ci ∈ [li, ui]
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60. Cardinality: A Filtering Algorithm
distribute([c1, , cm],[v1, , vm],[x1, ,xn]),
ci ∈ [li, ui]
transformed into a network flow problem
∃ solution iff ∃ feasible flow v1
[0,1]
[l1,u1] x1
[1,1] v2 T S xi vj xn
[lm,um] vm
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61. Cardinality: A Filtering Algorithm
Compute a maximum flow
Build the residual graph
Find its strongly connected components
Remove zero-flow arcs between components
achieves GAC
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62. Lexicographic Ordering: A Filtering Algorithm
A new family of global constraints
Linear time complexity
Ensures that a pair of vectors of variables are lexicographically ordered. 1 4 2
 lex 2 9 8 7
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63. Motivation: Symmetry
Symmetry: transformation of an entity that preserves the properties of the entity
Example:
180º
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64. Motivation: Symmetry Frequently occurs Tough for IP
Combinatorial problems like covering arrays
Rows and columns can be permuted
Messy real world problems like nurse rostering
Nurses with same skills can be swapped
Tough for IP
Very active research area within CP
Some effective techniques have been developed
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65. Motivation An important class of symmetries in CP
matrices of decision variables
rows/columns represent indistinguishable objects, hence symmetric
Rows and columns can be permuted without affecting satisfiability
Encountered frequently
Chapter 1, Section 1.2
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66. Example: Sports Scheduling
Schedule games between n teams over n-1 weeks
Each week is divided into n/2 periods
Each period has 2 slots: home and away
Find a schedule such that
every team plays exactly once a week
every team plays against every other team
every team plays at most twice in the same period over the tournament
Chapter 1, Section 1.2
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67. Example: Sport Scheduling
We need a table of meetings!
Period3
Period4
Period2
Period1
0 vs 7
2 vs 7
2 vs 6
0 vs 4
1 vs 6
3 vs 5
4 vs 5
0 vs 5
1 vs 4
3 vs 7
Week 5
3 vs 4
0 vs 6
1 vs 5
Week 6
1 vs 3
1 vs 2
2 vs 5
4 vs 6
6 vs 7
5 vs 6
5 vs 7
0 vs 3
1 vs 7
2 vs 3
2 vs 4
3 vs 6
4 vs 7
0 vs 2
0 vs 1
Week 7
Week 4
Week 3
Week 2
Week1
Chapter 1, Section 1.2
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68. Example: Sport Scheduling
We need a table of meetings!
Period3
Period4
Period2
Period1
0 vs 7
2 vs 7
2 vs 6
0 vs 4
1 vs 6
3 vs 5
4 vs 5
0 vs 5
1 vs 4
3 vs 7
Week 5
3 vs 4
0 vs 6
1 vs 5
Week 6
1 vs 3
1 vs 2
2 vs 5
4 vs 6
6 vs 7
5 vs 6
5 vs 7
0 vs 3
1 vs 7
2 vs 3
2 vs 4
3 vs 6
4 vs 7
0 vs 2
0 vs 1
Week 7
Week 4
Week 3
Week 2
Week1
Chapter 1, Section 1.2
Weeks are indistinguishable
Periods are indistinguishable
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69. Example: Sport Scheduling
Weeks are indistinguishable
Periods are indistinguishable
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Period 1
0 vs 1
0 vs 2
4 vs 7
3 vs 6
3 vs 7
1 vs 5
2 vs 4
Period 2
Chapter 1, Section 1.2
2 vs 3
1 vs 7
0 vs 3
5 vs 7
1 vs 4
0 vs 6
5 vs 6
Period 3
4 vs 5
3 vs 5
1 vs 6
0 vs 4
2 vs 6
2 vs 7
0 vs 7
Period 4
6 vs 7
4 vs 6
2 vs 5
1 vs 2
0 vs 5
3 vs 4
1 vs 3
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70. Example: Sport Scheduling
Weeks are indistinguishable
Periods are indistinguishable
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Period 1
0 vs 1
3 vs 7
4 vs 7
3 vs 6
0 vs 2
1 vs 5
2 vs 4
Period 2
Chapter 1, Section 1.2
2 vs 3
1 vs 4
0 vs 3
5 vs 7
1 vs 7
0 vs 6
5 vs 6
Period 3
4 vs 5
2 vs 6
1 vs 6
0 vs 4
3 vs 5
2 vs 7
0 vs 7
Period 4
6 vs 7
0 vs 5
2 vs 5
1 vs 2
4 vs 6
3 vs 4
1 vs 3
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71. Example: Bin Packing Consider 2 identical bins: A B 10/04/2017
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72. Example Consider 2 identical bins: We must pack 6 items: A B 1 2 3 4 5
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73. Example Here is one solution: 5 6 3 4 1 2 A B 10/04/2017
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74. Example Here is another: 6 5 4 3 2 1 A B 10/04/2017
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75. Example Is there any fundamental difference? 5 6 a) 3 4 1 2 A B 6 5 b)
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76. Example Consider a matrix model: 1 2 3 4 5 6 A B 1 2 3 4 5 6 A B 5 6B 5 6 a) 3 4 1 2 A B 1 2 3 4 5 6 A B 6 5 b) 4 3 2 1 A B
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77. Example Consider a matrix model: 1 2 3 4 5 6 A B 1 2 3 4 5 6 A B
NB: ‘1’ means place this item in this bin:
Consider a matrix model: 1 2 3 4 5 6 A B 5 6 a) 3 4 1 2 A B 1 2 3 4 5 6 A B 6 5 b) 4 3 2 1 A B
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78. Example Consider a matrix model: 1 2 3 4 5 6 A B 1 2 3 4 5 6 A BB
If we insist that row A  lex row B, we remove a) from the solution set. 1 2 3 4 5 6 A B 6 5 b) 4 3 2 1 A B
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79. Example Notice that items 3 and 4 are identical. 1 2 3 4 5 6 A B 1 2 3B 6 5 b) 4 3 2 1 A B 1 2 3 4 5 6 A B 6 5 c) 3 4 2 1
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80. Example Notice that items 3 and 4 are identical. 1 2 3 4 5 6 A B 1 2 3B 6 5 b) 4 3 2 1 A B 1 2 3 4 5 6 A B
If we insist that col 3  lex col 4, we remove c) from the solution set.
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81. Aims Main Goal Aims: we focus on lexicographic ordering constraints
Eliminate row and column symmetries effectively and efficiently.
Aims:
Investigate types of ordering constraints to break row and column symmetries.
Devise global constraints to easily pose and efficiently solve the ordering constraints.
Examine the effectiveness of the ordering constraint
we focus on lexicographic ordering constraints
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82. How GACLex Works Consider the following example.
We have two vectors of decision variables: x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1}
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83. How GACLex Works Consider the following example.
We have two vectors of decision variables: x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1}
We want to enforce GAC on: x lex y.
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84. A Tale of Two Pointers
We use two pointers, α and β, to avoid repeatedly traversing the vectors.
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85. A Tale of Two Pointers
We use two pointers, α and β, to avoid repeatedly traversing the vectors.
We index the vectors as follows: 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1}
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Most Significant Index
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86. A Tale of Two Pointers
We use two pointers, α and β, to avoid repeatedly traversing the vectors. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1}
α: index such that all variables at more significant indices are ground and equal.
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87. A Tale of Two Pointers
We use two pointers, α and β, to avoid repeatedly traversing the vectors. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1}
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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88. A Tale of Two Pointers
We use two pointers, α and β, to avoid repeatedly traversing the vectors. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1}
α: index such that all variables at more significant indices are ground and equal.
β: If tails never violate the constraint: 
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89. Pointer Initialisation
Needs one traversal of the vectors (linear). 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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90. Pointer Initialisation
Needs one traversal of the vectors (linear). 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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91. Failure Inconsistent if β  α.
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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92. How GACLex Works We maintain α and β as assignments made.
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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93. How GACLex Works We maintain α and β as assignments made.
When β = α + 1 we enforce bounds consistency on: xα < yα
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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94. How GACLex Works We maintain α and β as assignments made.
When β = α + 1 we enforce bounds consistency on: xα < yα
The variable at the αth element of each vector.
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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95. How GACLex Works We maintain α and β as assignments made.
When β = α + 1 we enforce bounds consistency on: xα < yα
When β > α + 1 we enforce bounds consistency on: xα  yα
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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96. How GACLex Works We maintain α and β as assignments made.
Key: we reduce GAC on vectors to BC on binary constraints.
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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97. How GACLex Works
0, 1 removed from yα. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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98. How GACLex Works
0, 1 removed from yα. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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99. How GACLex Works Update α.1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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100. How GACLex Works
Update α. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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101. How GACLex Works 3, 4 removed from xα. α β1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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102. How GACLex Works 3, 4 removed from xα. α β1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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103. How GACLex Works Update α. α β1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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104. How GACLex Works Update α. α β1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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105. How GACLex Works
4, 5 removed from xα, 0, 1 removed from yα. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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106. How GACLex Works
4, 5 removed from xα, 0, 1 removed from yα. 1 2 3 4 x
{2}
{1,3,4}
{1,2,3,4,5}
{1,2}
{3,4,5} y
{0,1,2}
{1}
{0,1,2,3,4}
{0,1} α β
α: index such that all variables at more significant indices are ground and equal.
β: most significant index from which the two vectors’ tails necessarily violate the constraint.
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107. Complexity Initialisation: O(n) Propagation:
We enforce bounds consistency between at most n pairs of variables: xα < yα or xα  yα.
Cost: b.
Overall cost: O(nb).
Amortised cost: O(b)
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108. Results: BIBD Decomposition takes
About 9 times longer on each of these instances.
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109. Constraint Propagation
General principle
Consistency
Filtering on simple constraints
Filtering on global constraints
Conclusion
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110. Conclusion Constraint propagation is the glue to combine efficient
filtering algorithms for common substructures
Matching theory, network flow theory, automata theory, computational geometry, …, encapsulated in constraints
Characterization of level of consistency
Aim for incrementality
Amount/frequency of filtering vs processing time
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111. ANY QUESTIONS?

112. Search

113. Search General principle Variable selection heuristics
Value selection heuristics
Conclusion
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114. General principle … var int Solution[Nodes] in Values; solve{
forall(e in Edges)
abs(Solution[e.x] - Solution[e.y]) >1;
alldifferent(Solution); };
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115. General principle … var int Solution[Nodes] in Values; solve{ };
search { }
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116. General principle With a good search strategy…
var int Solution[Nodes] in Values;
solve{
forall(e in Edges)
abs(Solution[e.x] - Solution[e.y]) >1;
alldifferent(Solution); };
search { }
With a good search strategy
We can quickly find good solution
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117. Searching for a good solution
For any interesting problem, propagation alone is not enough
We typically proceed by tree search
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118. Solving CSP by Search Search tree: root: empty node
one variable per level
sucessors of a node: every value of the next level var F N S
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119. Searching for a good solution
Two main decisions to control search
choose a variable
choose a value from its domain
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120. Search General principle Variable selection heuristics
Value selection heuristics
Conclusion
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121. Variable Selection Heuristics
Has an impact on tree topology
static vs dynamic ordering
smallest-domain-first
first-fail principle: “To succeed, try first where you are most likely to fail.”
regret: favour variable with greatest difference in cost between two best values in domain
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122. Search General principle Variable selection heuristics
Value selection heuristics
Conclusion
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123. Value Selection Heuristics
Not as critical
Very problem-dependent
Alternative for large domains: domain splitting
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124. Search General principle Variable selection heuristics
Value selection heuristics
Conclusion
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125. Conclusion A lot of control over the search strategy
Many heuristics developed, some of them generic
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126. ANY QUESTIONS?

127. Lot-sizing under demand uncertainty

128. Demand Uncertainty in Supply Chain Networks
Inventories
Production
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Sales

129. Demand Uncertainty in Supply Chain Networks?
When to order?
How much to order?
Inventories
Production
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Sales

130. Demand Uncertainty in Supply Chain Networks?
When to order?
How much to order?
Inventories
Production
Demand
Uncertainty
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Sales

131. Demand Uncertainty in Supply Chain Networks?
When to order?
How much to order?
Inventories
Production
Demand
Uncertainty
work in collaboration
with Bell Labs Ireland
Sales
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132. Demand Uncertainty in Supply Chain Networks
Determining the optimal inventory control policy parameters is key to profitability for any company involved in distribution and/or production of goods
We developed a CP model to find the optimal dynamic (R,S) inventory policy parameters such that
the expected cost is minimized;
demand is stochastic, non-stationary; and
a minimum service level is required
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133. Results
State-of-the-art improvement for the stochastic non-stationary formulation of the lot-sizing problem
Real-world instances can be solved in few seconds
The strategy could be extended to deal with
Capacity constraints
Lead time uncertainty
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134. (Demo)
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135. Experimental results
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136. ANY QUESTIONS?

137. Successful CP (Machine) Scheduling
(whole book on constraint-based scheduling)
Sports Scheduling (e.g. NFL)
Rostering
Allocation (e.g. terminal gates to aircrafts)
Transportation (e.g. VRP, airline crew rotation)
Even pure problems like Maximum Clique
Production planning (Lot-sizing)
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138. Finding out more Talks: CP, CPAIOR, IJCAI, AAAI, ECAI, INFORMS, CORS
Papers: Lecture Notes in Computer Science (Springer), Constraints (Kluwer), AI journals, OR journals
Software: CHIP; ECLiPSe; ECLAIR; FaCiLe;
ILOG OPL, Solver; SISCtus Prolog, Choco,. . .
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139. Finding out more Books:
Apt, K., Principles of Constraint Programming, Cambridge University Press,
Baptiste, P., Le Pape, C., Nuijten, W.. Constraint-Based Scheduling, Kluwer Academic Publishers,2001.
Hooker, J., Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction, John Wiley & Sons, 2000.
Marriott, K., Stuckey, P.J., Programming with Constraints: An Introduction, MIT Press, 1998.
Constraint and Integer Programming: Toward a Unified Methodology, edited by M. Milano, Kluwer Academic Publishers, 2003.
Tsang, E., Foundations of Constraint Satisfaction, Academic Press, 1993.
Van Hentenryck, P., The OPL Optimization Programming Language, MIT Press, 1999.
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140. Acknowledgements
Some parts of this tutorial are adapted material from tutorials given by:
Gilles Pesant
Pedro Messeguer
Chris Beck
Most parts of the tutorial is work done in collaboration with:
Alan Frisch, Ian miguel, Zeynep Kiziltan, Toby Walsh, Armagan Tarim, Roberto Rossi, Steven Prestwich
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