Lecture 13:
Modeling in
Constraint Programming
© J. Christopher Beck 2008 1

Outline


Introduction

The Crystal Maze

Heuristics, Propagation, B&B
Modeling in CP
© J. Christopher Beck 2008 2

Readings


P Ch 5.5, D.2, D.3
B.M. Smith,
“Modelling”, The
Handbook of
Constraint
Programming,
2006.
© J. Christopher Beck 2008 3

Crystal Maze
You have
5 minutes!

Place the numbers 1 through 8 in the nodes such that:

Each number appears exactly once
– No connected
?
?
nodes have consecutive numbers
?
?
?
?
© J. Christopher Beck 2008
?
?
4

Modeling




Each node  a variable
{1, …, 8}  values in the domain of each variable
No consecutive numbers  a constraint

(v i
, v j
)  |v i
– v j
| > 1
All values used  all-different constraint
© J. Christopher Beck 2008 5

Heuristic Search
{1, 2, 3, 4, 5, 6, 7, 8}
?
?
?
?
?
© J. Christopher Beck 2008
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6

Inference/Propagation
?
{1, 2, 3, 4, 5, 6, 7, 8}
?
?
?
{1, 2, 3, 4, 5, 6, 7, 8}
© J. Christopher Beck 2008
?
?
7

Inference/Propagation
{3, 4, 5, 6} {3, 4, 5, 6}
{3, 4, 5, 6, 7} {2, 3, 4, 5, 6}
{3, 4, 5, 6} {3, 4, 5, 6}
© J. Christopher Beck 2008 8

Generic CP Algorithm
Start
Propagators
Solution?
Success
Dead-end?
Backtrack
Technique
Nothing to retract?
Make
Heuristic
Decision
Failure
© J. Christopher Beck 2008
Assert
Commitment
9

Questions?
With CP You Are …



Guaranteed to find a solution if one exists

If not you can prove there is no solution
But you need to have enough time!

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 10

The Core of CP


Modeling

How to represent the problem
Heuristic search


How to branch
How much effort to find a good branch

Inference/propagation

How much effort

Backtracking
© J. Christopher Beck 2008 11

Constraint Satisfaction
Problem (CSP)


Given:



V, a set of variables {v
0
, v
1
, …, v n
}
D, a set of domains {D
0
, D
1
, …, D n
}
C, a set of constraints {c
0
, c
1
, …, c m
}
Each constraint, c i c i
(v
0
, v
2
, v
4
, v
117 it constrains
, has a scope
, …), the variables that
© J. Christopher Beck 2008 12

Constraint Satisfaction
Problem (CSP)

A constraint, c i
, is a mapping from the elements of the Cartesian product of the domains of the variables in its scope to {T,F}
 c i
(v
(D
0
0
, v
X D
2
2
, v
4
, v
117
X D
4
, …) maps:
X D
117
X … )  {T,F}

A constraint is satisfied if the assignment of the variables in its scope are mapped to T
© J. Christopher Beck 2008 13

Constraint Satisfaction
Problem (CSP)

In a solution to a CSP:

 each variable is assigned a value from its domain: v i
= d i
, d i є D i each constraint is satisfied
© J. Christopher Beck 2008 14

Variables & Constraints


Variables can be anything

 colors, times, countries, … often integer valued or binary
Constraints are not limited to linear constraints

 extensional: list of acceptable valuecombinations intensional: mathematical expression
© J. Christopher Beck 2008 15

Examples





V = {v
1
, v
2
, …, v n
}
D i
= {1, 2, …, n} all-diff(v
1

, v
2
, …, v n
) each of the variables must take on a different value v
1 v
4
> v
2
= v
5
+ v
4
* v
6
© J. Christopher Beck 2008 16

n -Queens Problem

Place n -Queens on the chess board so that no queen can attack any other
© J. Christopher Beck 2008 17

n -Queens Problem


Variables?
Constraints?
© J. Christopher Beck 2008 18

n -Queens Problem

How about a different model?
© J. Christopher Beck 2008 19

Why Do We Care About
Modeling?


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 20