Constraint Programming I

1
Constraint Programming I

• March 13, 2001

Martin Henz, School of Computing,
NUS www.comp.nus.edu.sg/henz
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Today

• Constraint programming in a nutshell
• Constraint propagation
• Branching
• Exploration
• Other search components

3
Thursday

• Branching and Exploration in OPL
• Case study ACC 97/98 Basketball
• Constraint programming techniques

4
Today

• Constraint programming in a nutshell
• Propagation
• Branching
• Exploration
• Other search components

5
Constraint Programming in a Nutshell

• SEND MORE MONEY

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Constraint Programming in a Nutshell

• SEND MORE MONEY

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SEND MORE MONEY

• Assign distinct digits to the letters
• S, E, N, D, M, O, R, Y
• such that
• S E N D
• M O R E
• M O N E Y holds.

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SEND MORE MONEY

• Assign distinct digits to the letters
• S, E, N, D, M, O, R, Y
• such that S E N D
• M O R E
• M O N E Y
• holds.

Solution 9 5 6 7 1 0 8
5 1 0 6 5 2
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Modeling

• Formalize the problem as a constraint problem
• n variables
• m constraints c1,,cm ? ?n
• problem Find a (v1,,vn)? ?n such
• that a ? ci , for all 1 ? i ?
  m

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A Model for MONEY

• 8 variables S,E,N,D,M,O,R,Y
• 5 constraints
• c1 (S,E,N,D,M,O,R,Y)? ?8 0 ? S,,Y ? 9
• c2 (S,E,N,D,M,O,R,Y)? ?8
• 1000S 100E 10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y

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A Model for MONEY (continued)

• more constraints
• c3 (S,E,N,D,M,O,R,Y)? ?8 S ? 0
• c4 (S,E,N,D,M,O,R,Y)? ?8 M ? 0
• c5 (S,E,N,D,M,O,R,Y)? ?8 SY all different

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Solution for MONEY

• c1 (S,E,N,D,M,O,R,Y)? ?8 0?S,,Y?9
• c2 (S,E,N,D,M,O,R,Y)? ?8
• 1000S 100E 10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y
• c3 (S,E,N,D,M,O,R,Y)? ?8 S ? 0
• c4 (S,E,N,D,M,O,R,Y)? ?8 M ? 0
• c5 (S,E,N,D,M,O,R,Y)? ?8 SY all
  different
• Solution (9,5,6,7,1,0,8,2)? ?8

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Constraint Programming
Exploiting constraints during tree search

• Choose propagation algorithms
• all different wait for fixing
• sum interval consistency
• Choose branching algorithms
• first-fail
• Choose exploration algorithms
• depth-first search

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S E N D M O R E M O N E Y
S ? ? E ? ? N ? ? D ? ? M ? ? O ? ? R ? ? Y ? ?

• 0?S,,Y?9
• S ? 0
• M ? 0
• SY all different
• 1000S 100E
  10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y

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S E N D M O R E M O N E Y
Propagate
S ? 0..9 E ? 0..9 N ? 0..9 D ? 0..9 M ?
0..9 O ? 0..9 R ? 0..9 Y ? 0..9

• 0?S,,Y?9
• S ? 0
• M ? 0
• SY all different
• 1000S 100E
  10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y

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S E N D M O R E M O N E Y
Propagate
S ? 1..9 E ? 0..9 N ? 0..9 D ? 0..9 M ?
1..9 O ? 0..9 R ? 0..9 Y ? 0..9

• 0?S,,Y?9
• S ? 0
• M ? 0
• SY all different
• 1000S 100E
  10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y

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S E N D M O R E M O N E Y
Propagate
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8

• 0?S,,Y?9
• S ? 0
• M ? 0
• SY all different
• 1000S 100E
  10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y

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S E N D M O R E M O N E Y
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
Branching
E ? 4
E 4
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8

• 0?S,,Y?9
• S ? 0
• M ? 0
• SY all different
• 1000S 100E
  10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y

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S E N D M O R E M O N E Y
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
Propagate
E ? 4
E 4
S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8

• 0?S,,Y?9
• S ? 0
• M ? 0
• SY all different
• 1000S 100E
  10N D
• 1000M 100O 10R E
• 10000M 1000O 100N 10E Y

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S E N D M O R E M O N E Y
Branching
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E ? 4
E 4

S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 5
E ? 5
S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
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S E N D M O R E M O N E Y
Propagate
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E ? 4
E 4

S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 5
E ? 5
S ? 9 E ? 5 N ? 6 D ? 7 M ? 1 O ? 0 R
? 8 Y ? 2
S ? 9 E ? 6..7 N ? 7..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
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S E N D M O R E M O N E Y
Complete Search Tree
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 4
E ? 4

S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 5
E ? 5
S ? 9 E ? 5 N ? 6 D ? 7 M ? 1 O ? 0 R
? 8 Y ? 2
S ? 9 E ? 6..7 N ? 7..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E ? 6
E 6


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Using OPL Syntax
enum Letter S,E,N,D,M,O,R,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lR10 lE lM10000
lO1000 lN100 lE10 lY search
forall(i in Letter ordered by increasing
dsize(li)) tryall(v in 0..9)
li v ?All Solutions Execution ? Run
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The Art of Constraint Programming

• Choose model
• Choose propagation algorithms
• Choose branching algorithms
• Choose exploration algorithms

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Demo SEND MORE MONEY
Click here for MONEY
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Constraint Programming Systems
support constraint programming with high-level
constructs

• Constraint programming library
• ILOG Solver (Puget 1993)
• Constraint programming languages
• CONSTRAINTS (Steele, Sussman 1980)
• CHIP (Dincbas, Hentenryck, Simonis, Aggoun 1988)
• CLP(R) (Jaffar, Maher, Stuckey, Yap 1992)
• Oz (Smolka and others 1995)
• OPL (Hentenryck 1998)

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Today

• Constraint programming in a nutshell
• Propagation
• Branching
• Exploration
• Other search components

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Issues in Propagation

• Expressivity What kind of information can be
  expressed as propagators?
• Completeness What behavior can be expected from
  propagation?
• Efficiency How much computational resources does
  propagation consume?

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Completeness

• Main purpose of propagation is exposure of
  inconsistencies.
• Inconsistencies allow to prune the search tree.
• An implementation is complete with respect to a
  set of constraints, if inconsistencies between
  constraints are guaranteed to be exhibited.

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Completeness

• General arithmetic constraints are undecidable
  (Hilberts Tenth Problem).
• Completeness is not always possible.
• Example
• c1 n gt 2
• c2 an bn cn

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Basic Constraints vs. Propagators

• Basic constraints
• are conjunctions of constraints of the form
  x ? S, where S is a finite set of integers
• enjoy complete constraint solving
• Propagators
• can be arbitrarily expressive (arithmetic,
  symbolic)
• implementation typically fast but incomplete

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Current Domain

• Let c be basic constraint. For a given variable
  x,
• the maximal set S such that c ? x ? S is
• consistent is called the current domain of x in
  c,
• denoted domc(x).
• Example
• C x?1..10, y?9..20,
• x?8..12
• domc(y) 9..20, domc(x)8..10

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Domain vs. Interval Consistency

• Domain consistency Check all elements of the
  domains of all variables known to the propagator
• Interval consistency Check only the boundaries
  of the domains of the variables

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Domain Consistency

• Assume a constraint d over variables x and
  y.
• A basic constraint c is domain consistent in x
• with respect to d, if for every i ? domc(x)
• there is a j ? domc(y) such that (i,j) ? d.
• Example
• d x y 10
• c x ? 7,9, y ? 1..20

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Domain Consistency

• Assume a constraint d over variables x and
  y.
• A basic constraint c is domain consistent in x
• with respect to d, if for every i ? domc(x)
• there is a j ? domc(y) such that (i,j) ? d.
• Example
• d x y 10
• c x ? 7,9, y ? 1,3

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Interval Consistency

• Assume a constraint d over variables x and
  y.
• A basic constraint c is interval consistent in x
  with
• respect to d, if for imin(domc(x)) and for
• imax(domc(x)) there is a j ? domc(y)
• such that (i,j) ? d.
• Example
• d x y 10
• C x ? 7,9, y ? 1..20

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Interval Consistency

• Assume a constraint d over variables x and
  y.
• A basic constraint c is interval consistent in x
  with
• respect to d, if for imin(domc(x)) and for
• imax(domc(x)) there is a j ? domc(y)
• such that (i,j) ? d.
• Example
• d x y 10
• C x?7,9, y?1..3

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Domain and Interval Consistency

• OPL syntax for all-different
• interval consistency alldifferent(l)
• domain consistency alldifferent(l) domain

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Some Propagator Classes

• Symbolic propagators
• Arithmetic propagators
• Reification

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Symbolic PropagatorsExample The Element
Propagator

• OPL arrayi x
• Meaning x is the ith element of array
• Example int a1..4 5,6,7,8
• var int i,x in 0..9
• solve ai x
• domc(i)1,3 ? x ? 5,7
• domc(x)6,8 ? i ? 2,4

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Arithmetic Propagators

• General arithmetic equations
• i1x11x1m1 inxn1xnmn 0
• lt gt lt gt ltgt

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Reified Constraints

• Reflecting the validity of a constraint in a
• 0/1 variable
• Example Reified arithmetic equations
• x (i1x11x1m1 inxn1xnmn 0)
• lt gt lt gt ltgt

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Propagation vs Branching

• trade-off
• complex propagation algorithms incur
• fewer, but more expensive nodes in
  tree
• Example MONEY with

alldiff and sum only test fixed assignment
alldiff wait for fixed variables sum interval
cons.
alldiff and sum domain consistency
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Today

• Constraint programming in a nutshell
• Propagation
• Branching
• Exploration
• Other search components

45
Branching Algorithms

• Constraint programming systems come with
• libraries of predefined branching algorithms
• programming support for user-defined branching
  algorithms

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Branching for MONEY
enum Letter S,E,N,D,M,O,R,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lR10 lE lM10000
lO1000 lN100 lE10 lY search
forall(i in Letter ordered by
increasing dsize(li)) tryall(v in 0..9)
li v ?All Solutions Execution ?
Run
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Basic Choice Points

• try x lt y x gt y endtry

x lt y
x gt y
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Choice Point Sequences

• try x lt y x gt y endtry
• try z 1 z 2 endtry

x lt y
x gt y
z 1
z 2
z 1
z 2
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Abbreviation tryall

• tryall(i in 1..5) x i
• stands for
• try x1x2x3x4x5 endtry

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Abbreviation forall

• forall(i in 1..4)
• try ai0 ai1 endtry
• stands for
• try a1 0 a1 1 endtry
• try a2 0 a2 1 endtry
• try a3 0 a3 1 endtry
• try a4 0 a4 1 endtry

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Examples of Branching Algorithms

• Enumeration Choose variable, choose value
• naive enumeration
  choose variables and values in
  a fixed sequence
• first-fail enumeration
  choose a variable with minimal
  domain size
• Domain-splitting
• try x lt mid x gt mid endtry
• Task sequencing for scheduling

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Today

• Constraint programming in a nutshell
• Propagation
• Branching
• Exploration
• Other search components

53
Exploration for MONEY
enum Letter S,E,N,D,M,O,R,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lR10 lE lM10000
lO1000 lN100 lE10 lY search
forall(i in Letter ordered by
increasing dsize(li)) tryall(v in 0..9)
li v ?All Solutions Execution ?
Run
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Exploration

• Depth-first search (default in OPL)
• Best-first search
• Limited discrepancy search Harvey/Ginsberg 95
• user-defined explorations

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Specifying Exploration in OPL
enum Letter S,E,N,D,M,O,T,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lS10 lT lM10000
lO1000 lN100 lE10 lY search
LDSearch(4) forall(i in Letter ordered by
increasing dsize(li)) tryall(v in 0..9)
li v
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Today

• Constraint programming in a nutshell
• Propagation
• Branching
• Exploration
• Other search components

57
Other Search Components

• Optimization
• Interaction
• Visualization
• Search Limits

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Optimization

• SEND MOST MONEY

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Optimization in OPL
enum Letter S,E,N,D,M,O,T,Y var int lLetter
in 0..9 maximize money subject to money
lM10000lO1000lN100lE10lY
alldifferent(l) lS ltgt 0 lM ltgt 0
lS1000 lE100 lN10 lD
lM1000 lO100 lS10
lT lM10000 lO1000 lN100
lE10 lY search forall(i in Letter
ordered by increasing dsize(li)) tryall(v
in 0..9) li v ?All Solutions
Execution ? Run
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Interaction

• First-solution search
• All solution search
• Last solution search
• Search with user interaction

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Visualization

• Example Oz Explorer Schulte 1997.
• Oz Explorer combines
• visualization
• first/all solution / user interaction
• branch-and-bound optimization
• depth-first search

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Search Limits

• SEND MOST MONEY
• but dont spend more than 10 hours searching!!!

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Search Limits in OPL
enum Letter S,E,N,D,M,O,T,Y var int lLetter
in 0..9 maximize money subject to money
lM10000lO1000lN100lE10lY
alldifferent(l) lS ltgt 0 lM ltgt 0
lS1000 lE100 lN10 lD
lM1000 lO100 lS10
lT lM10000 lO1000 lN100
lE10 lY search timeLimit 36000
forall(i in Letter ordered by increasing
dsize(li)) tryall(v in 0..9)
li v ?All Solutions Execution ? Run
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Thursday

• Branching and Exploration in OPL
• Case study ACC 97/98 Basketball
• Constraint programming techniques