Lecture 13:
Modeling in
Constraint Programming
© J. Christopher Beck 2008
1
Outline

Introduction
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
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The Crystal Maze
Heuristics, Propagation, B&B
Modeling in CP
© J. Christopher Beck 2008
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Readings
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
P Ch 5.5, D.2, D.3
B.M. Smith,
“Modelling”, The
Handbook of
Constraint
Programming,
2006.
© J. Christopher Beck 2008
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You have
5 minutes!
Crystal Maze

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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© J. Christopher Beck 2008
?
?
?
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Modeling
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Each node  a variable
{1, …, 8}  values in the domain of
each variable
No consecutive numbers  a constraint
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(vi, vj)  |vi – vj| > 1
All values used  all-different
constraint
© J. Christopher Beck 2008
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Heuristic Search
{1, 2, 3, 4, 5, 6, 7, 8}
?
© J. Christopher Beck 2008
?
?
1?
8?
?
?
?
6
Inference/Propagation
{1, 2, 3, 4, 5, 6, 7, 8}
?
?
?
1?
8?
?
?
?
{1, 2, 3, 4, 5, 6, 7, 8}
© J. Christopher Beck 2008
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Inference/Propagation
{3, 4, 5, 6}
4?
{3, 4, 5, 6}
6?
{3, 4, 5, 6, 7}
7?
{2, 3, 4, 5, 6}
1?
8?
3?
5?
{3,
4,
5,
6}
© J. Christopher Beck 2008
{3, 4, 5, 6}
2?
8
Generic CP Algorithm
Start
Solution?
Propagators
Success
Dead-end?
Backtrack
Technique
Nothing to
retract?
Failure
© J. Christopher Beck 2008
Make
Heuristic
Decision
Assert
Commitment
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Questions?
With CP You Are …

Guaranteed to find a solution if one
exists
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But you need to have enough time!
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
If not you can prove there is no solution
On average, for many problems you can
find a solution within a reasonable time
CP is commercially successful for
scheduling and other applications
© J. Christopher Beck 2008
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The Core of CP

Modeling
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Heuristic search
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How to branch
How much effort to find a good branch
Inference/propagation
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How to represent the problem
How much effort
Backtracking
© J. Christopher Beck 2008
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Constraint Satisfaction
Problem (CSP)
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Given:
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V, a set of variables {v0, v1, …, vn}
D, a set of domains {D0, D1, …, Dn}
C, a set of constraints {c0, c1, …, cm}
Each constraint, ci, has a scope
ci(v0, v2, v4, v117, …), the variables that
it constrains
© J. Christopher Beck 2008
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Constraint Satisfaction
Problem (CSP)

A constraint, ci, is a mapping from the
elements of the Cartesian product of
the domains of the variables in its scope
to {T,F}

ci(v0, v2, v4, v117, …) maps:
(D0 X D2 X D4 X D117 X … )  {T,F}
A constraint is satisfied if the
assignment of the variables in its scope
are
mapped
to
T
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© J. Christopher Beck 2008

Constraint Satisfaction
Problem (CSP)
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In a solution to a CSP:
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
each variable is assigned a value from its
domain: vi = di, di є Di
each constraint is satisfied
© J. Christopher Beck 2008
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Variables & Constraints
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Variables can be anything
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colors, times, countries, …
often integer valued or binary
Constraints are not limited to linear
constraints
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extensional: list of acceptable valuecombinations
intensional: mathematical expression
© J. Christopher Beck 2008
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Examples
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V = {v1, v2, …, vn}
Di = {1, 2, …, n}
all-diff(v1, v2, …, vn)
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each of the variables must take on a
different value
v1 > v 2 + v 4
v4 = v5 * v6
© J. Christopher Beck 2008
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n-Queens Problem
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© J. Christopher Beck 2008
Place n-Queens on
the chess board so
that no queen can
attack any other
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n-Queens Problem
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© J. Christopher Beck 2008
Variables?
Constraints?
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n-Queens Problem
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© J. Christopher Beck 2008
How about a
different model?
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Why Do We Care About
Modeling?
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
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You need to represent your problem!
As in MIP/LP different models are
harder or easier to solve
We will return to CP modeling (in a big
way) in a few weeks
© J. Christopher Beck 2008
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