1

Finite Constraint DomainsFinite Constraint Domains

2

Finite Constraint DomainsFinite Constraint Domains

Constraint satisfaction problems (CSP) A backtracking solver Node and arc consistency Bounds consistency Generalized consistency Optimization for arithmetic CSPs

3

Finite Constraint DomainsFinite Constraint Domains

An important class of constraint domains Use to model constraint problems involving

choice: e.G. Scheduling, routing and timetabling

The greatest industrial impact of constraint programming has been on these problems

4

Constraint Satisfaction Constraint Satisfaction ProblemsProblems

A constraint satisfaction problem (CSP) consists of: A constraint C over variables x1,..., Xn A domain D which maps each variable xi to a

set of possible values d(xi) It is understood as the constraint

C x D x xn D xn 1 1( ) ( )

5

Map ColouringMap Colouring

A classic CSP is the problem of coloring a map so that no adjacent regions have the same color

WA

NT

SA

Q

NSW

V

T

Can the map of Australia be colored with 3 colors ?

WA NT WA SA NT SA

NT Q SA Q SA NSW

SA V Q NSW NSW V

D WA D NT D SA D Q

D NSW D V D T

red yellow blue

( ) ( ) ( ) ( )

( ) ( ) ( )

{ , , }

6

4-queens4-queens

Place 4 queens on a 4 x 4 chessboard so that none can take another.

Q1 Q2 Q3 Q4

1

2

3

4

Four variables Q1, Q2, Q3, Q4 representing the row of the queen in each column. Domain of each variable is {1,2,3,4}

One solution! -->

7

4-queens4-queens

The constraints:

Q Q Q Q Q Q

Q Q Q Q Q Q

Q Q Q Q Q Q

Q Q Q Q Q Q

Q Q Q Q Q Q

Q Q Q Q Q Q

1 2 1 3 1 4

2 3 2 4 3 4

1 2 1 1 3 2 1 4 3

2 3 1 2 4 2 3 4 1

1 2 1 1 3 2 1 4 3

2 3 1 2 4 2 3 4 1

Not on the same row

Not diagonally up

Not diagonally down

8

Smugglers KnapsackSmugglers Knapsack

Smuggler with knapsack with capacity 9, who needs to choose items to smuggle to make profit at least 30

object profit size

whiskey

perfume

cigarretes

15 4

10 3

7 2

4 3 2 9 15 10 7 30W P C W P C

What should be the domains of the variables?

9

Systematic Search MethodsSystematic Search Methods

exploring the solution space complete and sound efficiency issues

Backtracking (BT) Generate & Test (GT)

exploringindividual assignments

exploring subspace

X

Z

Y

10

Generate & TestGenerate & Test probably the most general problem solving method Algorithm:

generate labelling

test satisfaction

Drawbacks: Improvements:blind generator smart generator

--> local search

late discovery of testing within generatorinconsistencies --> backtracking

11

Backtracking (BT)Backtracking (BT)

incrementally extends a partial solution towards a complete solution

Algorithm:

assign value to variable

check consistency

until all variables labelled Drawbacks:

thrashing redundant work late detection of conflict

A

C

B

1

1

1

1

2

2

A = D, B D, A+C < 4

1

2

D21

1

12

GT & BT - ExampleGT & BT - Example

Problem:X::{1,2}, Y::{1,2}, Z::{1,2}

X = Y, X Z, Y > Z

generate & test backtracking

X Y Z test1 1 1 fail1 1 2 fail1 2 1 fail1 2 2 fail2 1 1 fail2 1 2 fail2 2 1 passed

X Y Z test1 1 1 fail

2 fail2 fail

2 1 fail2 1 passed

13

Simple Backtracking SolverSimple Backtracking Solver

The simplest way to solve CSPs is to enumerate the possible solutions

The backtracking solver: Enumerates values for one variable at a time Checks that no prim. Constraint is false at each stage

Assume satisfiable(c) returns false when primitive constraint c with no variables is unsatisfiable

14

Partial SatisfiablePartial Satisfiable

Check whether a constraint is unsatisfiable because of a prim. Constraint with no vars

Partial_satisfiablePartial_satisfiable(c) For each primitive constraint c in C

If vars(c) is empty If satisfiable(c) = false return false

Return true

15

Backtrack SolveBacktrack Solve

Back_solveBack_solve(c,d) If vars(c) is empty return partial_satisfiablepartial_satisfiable(c) Choose x in vars(c) For each value d in d(x)

Let C1 be C with x replaced by d If partial_satisfiable(partial_satisfiable(c1) then

If back_solveback_solve(c1,d) then return true Return false

16

Backtracking SolveBacktracking SolveX Y Y Z D X D Y D Z ( ) ( ) ( ) { , }1 2

X Y Y Z

Choose var X domain {1,2}

X 1

1 Y Y ZChoose var Y domain {1,2}

Y 1

1 1 1 Z

partial_satisfiablepartial_satisfiable false

Y 2

1 2 2 Z

Choose var Z domain {1,2}

Z 1

1 2 2 1 No variables, and false

Z 2

1 2 2 2

No variables, and false

Variable X domain {1,2}

X 2

2 Y Y ZChoose var Y domain {1,2}

2 1 1 Z 2 2 2 Z

17

Consistency TechniquesConsistency Techniques

removing inconsistent values from variables’ domains graph representation of the CSP

binary and unary constraints only (no problem!) nodes = variables edges = constraints

node consistency (NC) arc consistency (AC) path consistency (PC) (strong) k-consistency

A

B

C

A>5

AB

A<C

B=C

18

Node and Arc ConsistencyNode and Arc Consistency

Basic idea: find an equivalent CSP to the original one with smaller domains of vars

Key: examine 1 prim.Constraint c at a time Node consistency: (vars(c)={x}) remove any

values from domain of x that falsify c Arc consistency: (vars(c)={x,y}) remove

any values from d(x) for which there is no value in d(y) that satisfies c and vice versa

19

Node ConsistencyNode Consistency

Primitive constraint c is node consistent with domain D if |vars(c)| /=1 or If vars(c) = {x} then for each d in d(x) X assigned d is a solution of c

A CSP is node consistent if each prim. Constraint in it is node consistent

20

Node Consistency ExamplesNode Consistency Examples

Example CSP is not node consistent (see Z)X Y Y Z Z

D X D Y D Z

2

1 2 3 4( ) ( ) ( ) { , , , }

This CSP is node consistent

X Y Y Z Z

D X D Y D Z

2

1 2 3 4 1 2( ) ( ) { , , , }, ( ) { , }

The map coloring and 4-queens CSPs are node consistent. Why?

21

Achieving Node ConsistencyAchieving Node Consistency

Node_consistentNode_consistent(c,d) For each prim. Constraint c in C

D := node_consistent_primitivenode_consistent_primitive(c, D) Return D

Node_consistent_primitiveNode_consistent_primitive(c, D) If |vars(c)| =1 then

Let {x} = vars(c)

Return DD x d D x x d c( ): { ( )|{ } } is a solution of

22

Arc ConsistencyArc Consistency

A primitive constraint c is arc consistent with domain D if |vars{c}| != 2 or Vars(c) = {x,y} and for each d in d(x) there

exists e in d(y) such that

And similarly for y A CSP is arc consistent if each prim.

Constraint in it is arc consistent

{ , }x d y e c is a solution of

23

Arc Consistency ExamplesArc Consistency Examples

This CSP is node consistent but not arc consistent X Y Y Z Z

D X D Y D Z

2

1 2 3 4 1 2( ) ( ) { , , , }, ( ) { , }

For example the value 4 for X and X < Y.

The following equivalent CSP is arc consistent

X Y Y Z Z

D X D Y D Z

2

( ) ( ) ( )

The map coloring and 4-queens CSPs are also arc consistent.

24

Achieving Arc ConsistencyAchieving Arc Consistency

Arc_consistent_primitiveArc_consistent_primitive(c, D) If |vars(c)| = 2 then

Return D Removes values which are not arc consistent with

c

D x d D x e D y

x d y e c

( ): { ( )| ( ),

{ , } }

exists

is a soln of

D y e D y d D x

x d y e c

( ): { ( )| ( ),

{ , } }

exists

is a soln of

25

Achieving Arc ConsistencyAchieving Arc Consistency

Arc_consistentArc_consistent(c,d) Repeat

W := d For each prim. Constraint c in C

D := arc_consistent_primitivearc_consistent_primitive(c,d) Until W = D Return D

A very naive version (there are much better)

26

Using Node and Arc Cons.Using Node and Arc Cons.

We can build constraint solvers using the consistency methods

Two important kinds of domain False domain: some variable has empty

domain Valuation domain: each variable has a

singleton domain Extend satisfiable to CSP with val. Domain

27

Node and Arc Cons. SolverNode and Arc Cons. Solver

D := node_consistentnode_consistent(C,D) D := arc_consistentarc_consistent(C,D) if D is a false domain

return false if D is a valuation domain

return satisfiable(C,D) return unknown

28

Node and Arc Solver ExampleNode and Arc Solver ExampleColouring Australia: with constraints

WA NT WA SA NT SA

NT Q SA Q SA NSW

SA V Q NSW NSW V

WA red NT yellow

WA NT SA Q NSW V T

Node consistency

29

Node and Arc Solver ExampleNode and Arc Solver ExampleColouring Australia: with constraints

WA NT WA SA NT SA

NT Q SA Q SA NSW

SA V Q NSW NSW V

WA red NT yellow

WA NT SA Q NSW V T

Arc consistency

30

Node and Arc Solver ExampleNode and Arc Solver ExampleColouring Australia: with constraints

WA NT WA SA NT SA

NT Q SA Q SA NSW

SA V Q NSW NSW V

WA red NT yellow

WA NT SA Q NSW V T

Arc consistency

31

Node and Arc Solver ExampleNode and Arc Solver ExampleColouring Australia: with constraints

WA NT WA SA NT SA

NT Q SA Q SA NSW

SA V Q NSW NSW V

WA red NT yellow

WA NT SA Q NSW V T

Arc consistency

Answer:

unknown

32

Backtracking Cons. SolverBacktracking Cons. Solver

We can combine consistency with the backtracking solver

Apply node and arc consistency before starting the backtracking solver and after each variable is given a value

33

Back. Cons Solver ExampleBack. Cons Solver Example

There is no possible value for

variable Q3!

Q1 Q2 Q3 Q4

1

2

3

4

No value can be

assigned toQ3 in this

case!

Therefore,we need to

choose another value

for Q2.

34

Back. Cons Solver ExampleBack. Cons Solver ExampleQ1 Q2 Q3 Q4

1

2

3

4

We cannot find any

possible valuefor Q4 inthis case!

Backtracking…

Find another value for Q3?

No!

backtracking,

Find anothervalue of Q2?

No!

backtracking,

Find anothervalue of Q1?

Yes, Q1 = 2

35

Back. Cons Solver ExampleBack. Cons Solver ExampleQ1 Q2 Q3 Q4

1

2

3

4

36

Back. Cons Solver ExampleBack. Cons Solver ExampleQ1 Q2 Q3 Q4

1

2

3

4

37

Node and Arc Solver ExampleNode and Arc Solver ExampleColouring Australia: with constraints

WA red NT yellow

WA NT SA Q NSW V T

Backtracking enumeration

Select a variable with domain of more than 1, T

Add constraint T red Apply consistency

Answer: true

38

Is AC enough?Is AC enough?

empty domain => no solution cardinality of all domains is 1 => solution Problem:

X::{1,2}, Y::{1,2}, Z::{1,2} X Y, X Z, Y Z

1 2

1 2

1 2

X

YZ

39

Path Consistency (PC)Path Consistency (PC)

Path (V0 … Vn) is path consistent iff for each pair of compatible values x in D(V0) and y in D(Vn) there exists an instantiation of the variables V1 ..Vn-1 such that all the constraint (Vi,Vi+1) are satisfied

CSP is path consistent iff each path is path consistent

consistency along the path only

V0 V1

V3

V2 V4 V5

???

40

Path Consistency (PC)Path Consistency (PC)

checking paths of length 2 is enough Plus/Minus

+ detects more inconsistencies than AC

- extensional representation of constraints (01 matrix), huge memory consumption

- changes in graph connectivity

V0 V1

V3

V2 V4 V5

???

41

K -consistencyK -consistency

K-consistency consistent valuation o (K-1) variables can be

extended to K-th variable strong K-consistency

J-consistency for each JK

42

Is k-consistency enough ?Is k-consistency enough ?

If all the domains have cardinality 1=> solution If any domain is empty => no solution Otherwise ?

strongly k-consistent constraint graph with k nodes => completeness

for some special kind of graphs weaker forms of consistency are enough

43

Consistency CompletenessConsistency Completeness strongly N-consistent constraint graph with N nodes

=> solution strongly K-consistent constraint graph with N nodes

(K<N) => ???

path consistentbut no solution

Special graph structures tree structured graph => (D)AC is enough

A

B C

D

{1,2,3}

{1,2,3}

{1,2,3}

{1,2,3}

44

hyper-arc consistencyhyper-arc consistency

A primitive constraint c is hyper-arc consistent with domain D if

for each variable x in c and for each d in d(x) the valuation x-->d can be extended to a solution of c

A CSP is hyper-arc consistent if each prim. Constraint in it is hyper-arc consistent

NC hyper-arc consistency for constraint with 1 var AC hyper-arc consistency for constraint with 2 var

45

Non binary constraints Non binary constraints

What about prim. constraints with more than 2 variables?

Each CSP can be transformed into an equivalent binary CSP (dual encoding)

k-consitncy hyper-arc consistency

46

Dual encodingDual encoding

Each CSP can be transformed into an equivalent binary CSP by ``dual encoding’’:

k-ary constraint c is converted to a dual variable vc with the domain consisting of compatible tuples

for each pair of constraint c,c’ sharing some variable there is a binary constraint between vc and vc’ restricting the dual variables to tuples in which the original shared variables take the same value

47

Hyper-arc consistencyHyper-arc consistency

hyper-arc consistency: extending arc consistency to arbitrary number of variables

(better than binary representation) Unfortunately determining hyper-arc

consistency is NP-hard (as expensive as determinining if a CSP is satisfiable).

Solution?

48

Bounds ConsistencyBounds Consistency

arithmetic CSP: domains = integers range: [l..u] represents the set of integers {l,

l+1, ..., u} idea use real number consistency and only

examine the endpoints (upper and lower bounds) of the domain of each variable

Define min(D,x) as minimum element in domain of x, similarly for max(D,x)

49

Bounds ConsistencyBounds Consistency

A prim. constraint c is bounds consistent with domain D if for each var x in vars(c) exist real numbers d1, ..., dk for remaining vars

x1, ..., xk such that min(D,xj),= dj<= max(D,xj) and is a solution of c

and similarly for An arithmetic CSP is bounds consistent if all

its primitive constraints are

{ min( , ), , }x D x x d xk dk 1 1

{ max( , )}x D x

50

Bounds Consistency ExamplesBounds Consistency Examples

X Y Z

D X D Y D Z

3 5

2 7 0 2 1 2( ) [ .. ], ( ) [ .. ], ( ) [ .. ]

Not bounds consistent, consider Z=2, then X-3Y=10

But the domain below is bounds consistent

D X D Y D Z( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 2 7 0 2 0 1

Compare with the hyper-arc consistent domain

D X D Y D Z( ) { , , }, ( ) { , , }, ( ) { , } 3 5 6 0 1 2 0 1

51

Achieving Bounds Achieving Bounds ConsistencyConsistency

Given a current domain D we wish to modify the endpoints of domains so the result is bounds consistent

propagation rules do this

52

Achieving Bounds Achieving Bounds ConsistencyConsistency

Consider the primitive constraint X = Y + Z which is equivalent to the three forms

X Y Z Y X Z Z X Y

Reasoning about minimum and maximum values:X D Y D Z X D Y D Z

Y D X D Z Y D X D Z

Z D X D Y Z D X D Y

min( , ) min( , ) max( , ) max( , )

min( , ) max( , ) max( , ) min( , )

min( , ) max( , ) max( , ) min( , )

Propagation rules for the constraint X = Y + Z

53

Achieving Bounds Achieving Bounds ConsistencyConsistency

X Y Z

D X D Y D Z

( ) [ .. ], ( ) [ .. ], ( ) [ .. ]4 8 0 3 2 2

The propagation rules determine that:

( ) ( )

( ) ( )

( ) ( )

0 2 2 5 3 2

4 2 2 6 8 2

4 3 1 8 8 0

X

Y

Z

Hence the domains can be reduced to

D X D Y D Z( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 4 5 2 3 2 2

54

More propagation rulesMore propagation rules4 3 2 9

9

4

3

4

2

49

3

4

3

2

39

2

4

2

3

2

W P C

W D P D C

P D W D C

C D W D P

min( , ) min( , )

min( , ) min( , )

min( , ) min( , )

Given initial domain:D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 9 0 9 0 9

We determine that

new domain:

W P C 9

4

9

3

9

2, ,

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 2 0 3 0 4

55

Disequations Disequations

Disequations give weak propagation rules, only when one side takes a fixed value that equals the minimum or maximum of the other is there propagation

Y Z

D Y D Z

D Y D Z

D Y D Z D Y D Z

( ) [ .. ], ( ) [ .. ]

( ) [ .. ], ( ) [ .. ]

( ) [ .. ], ( ) [ .. ] ( ) [ .. ], ( ) [ .. ]

2 4 2 3

2 4 3 3

2 4 2 2 3 4 2 2

no propagation

no propagation

prop

56

MultiplicationMultiplication X Y Z

If all variables are positive its simple enoughX D Y D Z X D Y D Z

Y D X D Z Y D X D Z

Z D X D Y Z D X D Y

min( , ) min( , ) max( , ) max( , )

min( , ) / max( , ) max( , ) / min( , )

min( , ) / max( , ) max( , ) / min( , )

But what if variables can be 0 or negative?

Example:

becomes:

D X D Y D Z( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 4 8 1 2 1 3

D X D Y D Z( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 4 6 2 2 2 3

57

MultiplicationMultiplication X Y Z

Calculate X bounds by examining extreme values

X D Y D Z D Y D Z

D Y D Z D Y D Z

minimum{min( , ) min( , ), min( , ) max( , )

max( , ) min( , ), max( , ) max( , )}

Similarly for upper bound on X using maximum

BUT this does not work for Y and Z? As long as min(D,Z) <0 and max(D,Z)>0 there is no bounds restriction on Y

X Y Z X Y d Z d { , , / } 4 4

Recall we are using real numbers (e.g. 4/d)

58

MultiplicationMultiplication X Y Z

We can wait until the range of Z is non-negative or non-positive and then use rules like

Y D X D Z D X D Z

D X D Z D X D Z

minimum{min( , ) / min( , ), min( , ) / max( , )

max( , ) / min( , ), max( , ) / max( , )}

division by 0:

ve ve

0 0

0

0

59

Bounds Consistency AlgmBounds Consistency Algm

Repeatedly apply the propagation rules for each primitive constraint until there is no change in the domain

We do not need to examine a primitive constraint until the domains of the variables involve are modified

60

Bounds Consistency ExampleBounds Consistency Example

Smugglers knapsack problem (no whiskey available)

capacity profit

W P C W P C4 3 2 9 15 10 7 30

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 0 9 0 9

W P C 102 15 12 10 60 7/ / /W P C 9 4 9 3 9 2/ / /

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 0 3 0 4

W P C 28 15 2 10 0 7/ / /

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 3 0 4

Continuing there is no further change

Note how we had to reexamine the profit constraint

61

Bounds consistency solverBounds consistency solver

D := bounds_consistentbounds_consistent(C,D) if D is a false domain

return false if D is a valuation domain

return satisfiable(C,D) return unknown

62

Back. Bounds Cons. SolverBack. Bounds Cons. Solver

Apply bounds consistency before starting the backtracking solver and after each variable is given a value

63

Back. Bounds Solver ExampleBack. Bounds Solver Example

Smugglers knapsack problem (whiskey available)capacity profit

W P C W P C4 3 2 9 15 10 7 30

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 9 0 9 0 9Current domain:

Initial bounds consistency

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 2 0 3 0 4

W = 0

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 3 0 4

P = 1

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 1 3 3

(0,1,3)

Solution Found: return true

64

Back. Bounds Solver ExampleBack. Bounds Solver Example

Smugglers knapsack problem (whiskey available)capacity profit

W P C W P C4 3 2 9 15 10 7 30 Current domain:

Initial bounds consistencyBacktrack

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 3 0 4

P = 2

D W D P D C( ) , ( ) , ( )

false

Backtrack

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 3 0 4

P = 3

D W D P D C( ) { }, ( ) { }, ( ) { } 0 3 0

W = 0

P = 1

(0,1,3) (0,3,0)

W = 1

(1,1,1)

W = 2

(2,0,0)

No more solutions

65

Generalized ConsistencyGeneralized Consistency

Can use any consistency method with any other communicating through the domain, node consistency : prim constraints with 1 var arc consistency: prim constraints with 2 vars bounds consistency: other prim. constraints

Sometimes we can get more information by using complex constraints and special consistency methods

66

AlldifferentAlldifferent

alldifferent({V1,...,Vn}) holds when each variable V1,..,Vn takes a different value alldifferent({X, Y, Z}) is equivalent to

Arc consistent with domain

BUT there is no solution! specialized consistency for alldifferent can find it

X Y X Z Y Z

D X D Y D Z( ) { , }, ( ) { , }, ( ) { , } 1 2 1 2 1 2

67

Alldifferent ConsistencyAlldifferent Consistency

let c be of the form alldifferent(V) while exists v in V where D(v) = {d}

V := V - {v} for each v’ in V

D(v’) := D(v’) - {d} DV := union of all D(v) for v in V if |DV| < |V| then return false domain return D

68

Alldifferent ExamplesAlldifferent Examplesalldifferent X Y Z

D X D Y D Z

({ , , })

( ) { , }, ( ) { , }, ( ) { , } 1 2 1 2 1 2

DV = {1,2}, V={X,Y,Z} hence detect unsatisfiabilityalldifferent X Y Z T

D X D Y D Z D T

({ , , , })

( ) { , }, ( ) { , }, ( ) { , }, ( ) { , , , } 1 2 1 2 1 2 2 3 4 5

DV = {1,2,3,4,5}, V={X,Y,Z,T} don’t detect unsat. Maximal matching based consistency could

X Y Z T

1 2 3 4 5

69

Other Complex ConstraintsOther Complex Constraints

schedule n tasks with start times Si and durations Di needing resources Ri where L resources are available at each moment

array access if I = i, then X = Vi and if X != Vi then I != i

cumulative S S D D R R Ln n n([ , , ],[ , , ],[ , , ], )1 1 1

element I V V Xn( ,[ , , ], )1

70

Optimization for CSPsOptimization for CSPs

Because domains are finite can use a solver to build a straightforward optimizer

retry_int_optretry_int_opt(C, D, f, best) D2 := int_solvint_solv(C,D) if D2 is a false domain then return best let sol be the solution corresponding to D2 return retry_int_optretry_int_opt(C /\ f < sol(f), D, f, sol)

71

Retry Optimization ExampleRetry Optimization Example

Smugglers knapsack problem (optimize profit)minimize subject to

15 10 7

4 3 2 9 15 10 7 30

0 9 0 9 0 9

W P Ccapacity profit

W P C W P C

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ]

First solution found: D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 1 3 3

Corresponding solution sol W P C{ , , } 0 1 3

sol f( ) 31

minimize subject to

15 10 7

4 3 2 9 15 10 7 30

15 10 7 31

0 9 0 9 0 9

W P Ccapacity profit

W P C W P C

W P C

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ]

Next solution found: D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 1 1 1 1 1 1

sol W P C{ , , } 1 1 1

sol f( ) 32

minimize subject to

15 10 7

4 3 2 9 15 10 7 30

15 10 7 31 15 10 7 32

0 9 0 9 0 9

W P Ccapacity profit

W P C W P C

W P C W P C

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ]

No next solution!

Return best solution

72

Backtracking OptimizationBacktracking Optimization

Since the solver may use backtrack search anyway combine it with the optimization

At each step in backtracking search, if best is the best solution so far add the constraint f < best(f)

73

Back. Optimization ExampleBack. Optimization Example

Smugglers knapsack problem (whiskey available)capacity profit

W P C W P C4 3 2 9 15 10 7 30

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 9 0 9 0 9Current domain:

Initial bounds consistency

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 2 0 3 0 4

W = 0

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 3 0 4

P = 1

D W D P D C( ) [ .. ], ( ) [ .. ], ( ) [ .. ] 0 0 1 1 3 3

(0,1,3)

Solution Found: add constraint

Smugglers knapsack problem (whiskey available)capacity profit

W P C W P C

W P C

4 3 2 9 15 10 7 30

15 10 7 31

74

Back. Optimization ExampleBack. Optimization Example

Initial bounds consistency

capacity profit

W P C W P C

W P C

4 3 2 9 15 10 7 30

15 10 7 31

P = 2

false

P = 3

false

W = 1

(1,1,1)

Modify constraint

capacity profit

W P C W P C

W P C

4 3 2 9 15 10 7 30

15 10 7 32

W = 0

P = 1

(0,1,3)

Smugglers knapsack problem (whiskey available)

W = 2

false

Return last sol (1,1,1)

75

Branch and Bound Opt.Branch and Bound Opt.

The previous methods,unlike simplex don't use the objective function to direct search

branch and bound optimization for (C,f) use simplex to find a real optimal, if solution is integer stop otherwise choose a var x with non-integer opt value

d and examine the problems

use the current best solution to constrain prob. ( , ) ( , )C x d f C x d f

76

Branch and Bound ExampleBranch and Bound ExampleSmugglers knapsack problem

{ / , , }W P C 11 4 0 0W 2 W 3

{ , / , }W P C 2 1 3 0P 0 P 1

{ , , / }W P C 2 0 1 2C 0 C 1

{ , , }W P C 2 0 0Solution (2,0,0) = 30

{ / , , }W P C 7 4 0 1W 1 W 2

{ , , / }W P C 1 0 5 2C 2 C 3

false { / , , }W P C 3 4 0 3W 0 W 1

false false

false

{ / , , }W P C 6 4 1 0W 1 W 2

{ , / , }W P C 1 5 3 0P 1 P 2

{ , , }W P C 1 1 1Solution (1,1,1) = 32

Worse than best sol

false

false

77

Finite Constraint Domains Finite Constraint Domains SummarySummary

CSPs form an important class of problems Solving of CSPs is essentially based on

backtracking search Reduce the search using consistency methods

node, arc, bound, generalized Optimization is based on repeated solving or

using a real optimizer to guide the search