The Theory and Practice
of Constraint Programming:
An Overview
Brahim Hnich
Faculty of Computer Science
Izmir University of Economics
Izmir, Turkey
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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Caveat
•In this talk:
Constraint programming for combinatorial problems
“Programming” refers to its roots in computer
science (programming languages)
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Outline
•Modelling
•Constraint propagation
•Search
•Demo: Lot-sizing with stochastic non-stationary
demand
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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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?
?
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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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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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Heuristic Search
{1, 2, 3, 4, 5, 6, 7, 8}
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1?
8?
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Inference/Propagation
{1, 2, 3, 4, 5, 6, 7, 8}
?
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1?
8?
?
?
?
{1, 2, 3, 4, 5, 6, 7, 8}
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Inference/Propagation
{3, 4, 5, 6}
{3, 4, 5, 6}
3?
6?
{3, 4, 5, 6, 7}
7?
{2, 3, 4, 5, 6}
1?
8?
4?
5?
{3, 4, 5, 6}
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2?
{3, 4, 5, 6}
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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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The Core of Constraint Computation
Modelling
Solving
Heuristic
Search
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Propagation
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ANY QUESTIONS?
Modeling
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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Constraint Satisfaction Problems
CSP: (X, D, C)
o X = {x1, x2,…, xn}
variables
o D = {d1, d2,…,dn}
domains (finite)
o C = {c1,c2,…,ce }
constraints
c ЄC
var(c) = {xi, xj,…, xk}
scope
rel(c) ⊆ di x dj x .. x dk permitted
tuples
Solution:
o assignment satisfying every constraint
o NP-complete task
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CSP: Relevance
Many problems can be represented as CSP:
Artificial Intelligence
Real-life
applications
o temporal reasoning
Control Theory
o controllers for sensory based robots
Concurency
Production planning
Staff scheduling
o process comm. and synchr.
Resource allocation
Computer Graphics
o geometric coherence
Circuit design
Database Systems
Option trading
o constraint databases
Bioinformatics
DNA sequencing
o sequence alignment
...
Operations research
o optimization
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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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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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Map Colouring
Variables: F, N, S
Values: {
}
Constraints: N ≠ S ≠ F ≠ N
S
F
N
A solution: F
N
S
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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:
1. Each word in S has 4 symbols from { C,G };
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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:
1. Each word in S has 4 symbols from { C,G };
2. Each pair of distinct words in S differ in at least 4
positions; and
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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:
1. Each word in S has 4 symbols from { C,G };
2. Each pair of distinct words in S differ in at least 4
positions; and
3. 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.
o
(x1,…,x8)R = ( x8,…,x1 ) is the reverse of ( x1,…,x8 )
o
(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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A Solution
S= { AAGCCGTT, TACGCGAT}
Each word in S has 4 symbols from { C,G };
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A Solution
S= { AAGCCGTT, TACGCGAT}
Each pair of distinct words in S differ
in at least 4 positions
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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 A4CP positions
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A Matrix Model
M
1
2
3
4
5
6
7
8
1
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ 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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A Matrix Model
1. Expressing that each word in S has 4 symbols from {
C,G }
M
1
2
3
4
5
6
7
8
1
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
…
m
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A Matrix Model
M
1
2
3
4
5
6
7
8
1
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ 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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A Matrix Model
1. …
2. Each pair of distinct words in S differ in at least 4
positions
M
1
2
3
4
5
6
7
8
1
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ 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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A Matrix Model
1. …
2. …
3. xR and yC differ in at least 4 positions.
MC
1
2
3
4
5
6
7
8
1
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ 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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A Matrix Model
1. …
2. …
3. xR and yC differ in at least 4 positions.
MC
1
2
3
4
5
6
7
8
1
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ A,C,G,T }
{ 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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ANY QUESTIONS?
Constraint Propagation
Constraint Propagation
•General principle
•Consistency
•Filtering on simple constraints
•Filtering on global constraints
•Conclusion
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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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General Principle
{1, 2, 3, 4, 5, 6, 7, 8}
Constraint network
≠
?
≠
?
{1, 2, 3, 4, 5, 6, 7, 8}
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≠
≠
≠
≠
≠
≠
1?
≠
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≠
≠
≠
≠
8?
≠
≠
≠
?
≠
?
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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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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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Constraint Propagation
•General principle
•Consistency
•Filtering on simple constraints
•Filtering on global constraints
•Conclusion
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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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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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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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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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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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Constraint Propagation
•General principle
•Consistency
•Filtering on simple constraints
•Filtering on global constraints
•Conclusion
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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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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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Constraint Propagation
•General principle
•Consistency
•Filtering on simple constraints
•Filtering on global constraints
•Conclusion
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Example: all-different
Var: F, N, S;
Val: {
F {
≠
Ctrs: N ≠ S ≠ F ≠ N
}
≠
N
{
};
}
≠
S
{
}
3 binary constraints,
they are GAC,
no pruning
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Example: all-different
Var: F, N, S;
Val: {
F {
≠
}
}
≠
Ctrs: N ≠ S ≠ F ≠ N
•Using binary disequalities,
this inconsistency goes
undetected
≠
N
{
};
S
{
}
3 binary constraints,
they are GAC,
no pruning
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Example: all-different
Var: F, N, S;
Val: {
F {
≠
N
{
}
≠
}
Ctrs: N ≠ S ≠ F ≠ N
We can do something simple:
≠
1. Count the number of
variables, n
{
2. Count the number of values
in the union of their domains,
m
S
3 binary constraints,
they are GAC,
no pruning
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};
}
3. If n > m then no solution can
possibly be found
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Example: all-different
Var: F, N, S;
Val: {
F {
≠
}
}
≠
S
{
3 binary constraints,
they are GAC,
no pruning
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Ctrs: N ≠ S ≠ F ≠ N
We can do something simple:
1. Count the number of variables, n
2. Count the number of values in the
union of their domains, m
3. If n > m then no solution can
possibly be found
≠
N
{
};
}
• ...still fooled by
Dx = Dy = Dz = {a, b}, Dw = {c, d}
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Example: all-different
Var: F, N, S;
Val: {
F {
≠
};
Ctrs: N ≠ S ≠ F ≠ N
}
F {
}
logically
equivalent
≠
all-different
N
{
}
≠
S
{
3 binary constraints,
they are GAC,
no pruning
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S
N
}
{
}
{
}
1 ternary constraint, not GAC,
GAC pruning  empty domain
no solution!!
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All-different: A Filtering Algorithm
•Build the corresponding bipartite graph
x1
1
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x2
2
x3
3
x4
4
x5
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6
7
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All-different: A Filtering Algorithm
•Build the corresponding bipartite graph
x1
1
x2
2
x3
3
x4
4
x5
5
x6
6
7
•∃ solution iff ∃ matching covering all the
variables
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All-different: A Filtering Algorithm
•Build the corresponding bipartite graph
x1
1
x2
2
x3
3
x4
4
x5
5
x6
6
7
•∃ 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]
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All-different: A Filtering Algorithm
•Build the corresponding bipartite graph
x1
1
x2
2
x3
3
x4
4
x5
5
x6
6
7
•∃ solution iff ∃ matching covering all the
variables
•starting from the current matching at each call
makes the algorithm incremental
•achieves GAC
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Cardinality: A Filtering Algorithm
•distribute([c1, . . . , cm],[v1, . . . , vm],[x1, . . . ,xn]),
ci ∈ [li, ui]
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Cardinality: A Filtering Algorithm
•distribute([c1, . . . , cm],[v1, . . . , vm],[x1, . . . ,xn]),
ci ∈ [li, ui]
•transformed into a network flow problem
•∃ solution iff ∃ feasible flow
[0,1]
v1
[l1,u1]
x1
[1,1]
S
v2
T
xi
vj
[lm,um]
xn
vm
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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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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.
0
1
4
2
8
7

2
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lex
9
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Motivation: Symmetry
Symmetry: transformation of an entity that
preserves the properties of the entity
Example:
180º
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Motivation: Symmetry
• Frequently occurs
 Combinatorial problems like
covering arrays
o Rows and columns can be permuted
 Messy real world problems
like nurse rostering
o 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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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
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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
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Example: Sport Scheduling
• We need a table of meetings!
Week1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Period1
0 vs 1
0 vs 2
4 vs 7
3 vs 6
3 vs 7
1 vs 5
2 vs 4
Period2
2 vs 3
1 vs 7
0 vs 3
5 vs 7
1 vs 4
0 vs 6
5 vs 6
Period3
4 vs 5
3 vs 5
1 vs 6
0 vs 4
2 vs 6
2 vs 7
0 vs 7
Period4
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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Example: Sport Scheduling
• We need a table of meetings!
Week1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Period1
0 vs 1
0 vs 2
4 vs 7
3 vs 6
3 vs 7
1 vs 5
2 vs 4
Period2
2 vs 3
1 vs 7
0 vs 3
5 vs 7
1 vs 4
0 vs 6
5 vs 6
Period3
4 vs 5
3 vs 5
1 vs 6
0 vs 4
2 vs 6
2 vs 7
0 vs 7
Period4
6 vs 7
4 vs 6
2 vs 5
1 vs 2
0 vs 5
3 vs 4
1 vs 3
• Weeks are indistinguishable
• Periods are indistinguishable
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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
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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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
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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Example: Bin Packing
• Consider 2 identical bins:
A
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B
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Example
• Consider 2 identical bins:
A
B
• We must pack 6 items:
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1
2
3
4
5
6
72
Example
• Here is one solution:
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5
6
3
4
1
2
A
B
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Example
• Here is another:
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6
5
4
3
2
1
A
B
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Example
• Is there any fundamental difference?
a)
b)
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5
3
1
A
6
4
2
B
6
4
2
A
5
3
1
B
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Example
• Consider a matrix model:
a)
b)
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5
3
1
A
6
4
2
A
1 2 3 4 5 6
6
4
2
B
5
3
1
B
A 1 0 1 0 1 0
B 0 1 0 1 0 1
1 2 3 4 5 6
A 0 1 0 1 0 1
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B 1 0 1 0 1 0
76
Example
• Consider a matrix model:
a)
b)
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5
3
1
A
6
4
2
A
6
4
2
B
5
3
1
B
NB: ‘1’ means place
this item in this bin:
1 2 3 4 5 6
A 1 0 1 0 1 0
B 0 1 0 1 0 1
1 2 3 4 5 6
A 0 1 0 1 0 1
B 1 0 1 0 1 0
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Example
• Consider a matrix model:
If we insist that row A  lex row B,
we remove a) from the solution set.
1 2 3 4 5 6
A 1 0 1 0 1 0
B 0 1 0 1 0 1
b)
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6
4
2
A
5
3
1
B
1 2 3 4 5 6
A 0 1 0 1 0 1
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B 1 0 1 0 1 0
78
Example
• Notice that items 3 and 4 are identical.
b)
c)
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6
4
2
A
6
3
2
1 2 3 4 5 6
5
3
1
B
A 0 1 0 1 0 1
B 1 0 1 0 1 0
1 2 3 4 5 6
5
4
1
A 0 1 1 0 0 1
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B 1 0 0 1 1 0
79
Example
• Notice that items 3 and 4 are identical.
b)
6
4
2
A
1 2 3 4 5 6
5
3
1
B
A 0 1 0 1 0 1
B 1 0 1 0 1 0
If we insist that col 3  lex col 4,
we remove c) from the solution set.
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1 2 3 4 5 6
A 0 1 1 0 0 1
B 1 0 0 1 1 0
80
Aims
• Main Goal
 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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How GACLex Works
• Consider the following example.
• We have two vectors of decision variables:
x {2}
{1,3,4} {1,2,3,4,5}
y {0,1,2} {1}
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{0,1,2,3,4}
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{1,2} {3,4,5}
{0,1} {0,1,2}
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}
y {0,1,2} {1}
{0,1,2,3,4}
{1,2} {3,4,5}
{0,1} {0,1,2}
• We want to enforce GAC on: x lex y.
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A Tale of Two Pointers
• We use two pointers, α and β, to avoid
repeatedly traversing the vectors.
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A Tale of Two Pointers
• We use two pointers, α and β, to avoid
repeatedly traversing the vectors.
• We index the vectors as follows:
0
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}
{0,1,2}
Most Significant Index
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A Tale of Two Pointers
• We use two pointers, α and β, to avoid repeatedly
traversing the vectors.
0
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}
{0,1,2}
• α: index such that all variables at more significant
indices are ground and equal.
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A Tale of Two Pointers
• We use two pointers, α and β, to avoid
repeatedly traversing the vectors.
0
•
•
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}
{0,1,2}
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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A Tale of Two Pointers
• We use two pointers, α and β, to avoid repeatedly
traversing the vectors.
0
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}
{0,1,2}
•
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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Pointer Initialisation
• Needs one traversal of the vectors (linear).
0
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}
{0,1,2}
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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Pointer Initialisation
• Needs one traversal of the vectors (linear).
0
1
2
•
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}
{0,1,2}
α
•
3
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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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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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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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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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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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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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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How GACLex Works
• 0, 1 removed from yα.
0
1
2
•
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}
{0,1,2}
α
•
3
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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How GACLex Works
• 0, 1 removed from yα.
0
1
2
•
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}
{0,1,2}
α
•
3
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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How GACLex Works
• Update α.
0
1
2
•
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}
{0,1,2}
α
•
3
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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How GACLex Works
• Update α.
0
1
2
•
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}
{0,1,2}
α
•
3
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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How GACLex Works
• 3, 4 removed from xα.
0
x {2}
1
{1,3,4}
y {0,1,2} {1}
2
•
4
{1,2,3,4,5}
{1,2} {3,4,5}
{0,1,2,3,4}
{0,1} {0,1,2}
α
•
3
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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How GACLex Works
• 3, 4 removed from xα.
0
x {2}
1
{1,3,4}
y {0,1,2} {1}
2
•
4
{1,2,3,4,5}
{1,2} {3,4,5}
{0,1,2,3,4}
{0,1} {0,1,2}
α
•
3
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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How GACLex Works
• Update α.
0
x {2}
1
{1,3,4}
y {0,1,2} {1}
2
•
4
{1,2,3,4,5}
{1,2} {3,4,5}
{0,1,2,3,4}
{0,1} {0,1,2}
α
•
3
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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How GACLex Works
• Update α.
0
x {2}
1
{1,3,4}
y {0,1,2} {1}
2
•
4
{1,2,3,4,5}
{1,2} {3,4,5}
{0,1,2,3,4}
{0,1} {0,1,2}
α
•
3
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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How GACLex Works
• 4, 5 removed from xα, 0, 1 removed from yα.
0
x {2}
1
{1,3,4}
y {0,1,2} {1}
2
•
4
{1,2,3,4,5}
{1,2} {3,4,5}
{0,1,2,3,4}
{0,1} {0,1,2}
α
•
3
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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How GACLex Works
• 4, 5 removed from xα, 0, 1 removed from yα.
0
x {2}
1
{1,3,4}
y {0,1,2} {1}
2
•
4
{1,2,3,4,5}
{1,2} {3,4,5}
{0,1,2,3,4}
{0,1} {0,1,2}
α
•
3
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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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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Results: BIBD
Time(s)
10000
Decomposition takes
About 9 times
longer on each of
these instances.
Decomposition
1000
100
10
1
1
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10
100
GACLex
1000
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10000
108
Constraint Propagation
•General principle
•Consistency
•Filtering on simple constraints
•Filtering on global constraints
•Conclusion
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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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ANY QUESTIONS?
Search
Search
•General principle
•Variable selection heuristics
•Value selection heuristics
•Conclusion
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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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General principle
…
var int Solution[Nodes] in Values;
solve{
…
};
search {
…
}
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General principle
…
var int Solution[Nodes] in Values;
solve{
forall(e in Edges)
abs(Solution[e.x] - Solution[e.y]) >1;
alldifferent(Solution);
};
search {
…
}
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With a good search strategy
We can quickly find good solution
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Searching for a good solution
•For any interesting problem, propagation alone
is not enough
•We typically proceed by tree search
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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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Searching for a good solution
•Two main decisions to control search
choose a variable
choose a value from its domain
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Search
•General principle
•Variable selection heuristics
•Value selection heuristics
•Conclusion
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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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Search
•General principle
•Variable selection heuristics
•Value selection heuristics
•Conclusion
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Value Selection Heuristics
•Not as critical
•Very problem-dependent
•Alternative for large domains: domain splitting
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Search
•General principle
•Variable selection heuristics
•Value selection heuristics
•Conclusion
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Conclusion
•A lot of control over the search strategy
•Many heuristics developed, some of them
generic
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ANY QUESTIONS?
Lot-sizing under demand uncertainty
Demand Uncertainty in Supply Chain Networks
Inventories
Production
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Sales
128
Demand Uncertainty in Supply Chain Networks
?
When to order?
How much to order?
Inventories
Production
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129
Demand Uncertainty in Supply Chain Networks
?
When to order?
How much to order?
Inventories
Production
Demand
Uncertainty
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Sales
130
Demand Uncertainty in Supply Chain Networks
?
When to order?
How much to order?
Inventories
Production
Demand
work in collaboration
Uncertainty
with Bell Labs Ireland
Sales
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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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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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(Demo)
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Experimental results
Seasonal Demands
100000
10000
seconds
1000
100
ILOG - CP model
10
CPLEX - MIP model
Choco - CP model dyn red
1
0.1
24 26 28 30 32 34 36 38 40 42 44 46 48 50
0.01
0.001
periods
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ANY QUESTIONS?
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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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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Finding out more
Books:
•Apt, K., Principles of Constraint Programming, Cambridge University Press,
2003.
•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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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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