general principles of constraint programming

1

General Principles of Constraint Programming

Jean-Charles REGINDirector of Constraint Programming

ILOG, Sophia Antipolis, [email protected]

2

Plan

General Principles CP vs other techniques Filtering algorithms An example: sports scheduling Strength of CP Weakness of CP New research Area Recent advances in CP at ILOG Conclusion

3

History

General Problem Solvers in 70’s ALICE [J-F Lauriere, AIJ 78], phD in Paris VI, 76

Prolog: CHIP, ECRC Munich 86 (Alice in Prolog) Colmerauer, Gallaire, Van Hentenryck

Constraint Satisfaction Problems Waltz 72, Mackworth 74, Freuder 76

Industry: Bull (Charme), ILOG (Solver) 92

4

Constraint Programming

In CP a problem is defined from:- variables with possible values (domain)- constraints

Domain can be discrete or continuous, symbolic values or numerical values

Constraints express properties that have to be satisfied

5

Problem = conjunction of sub-problems

In CP a problem can be viewed as a conjunction of sub-problems that we are able to solve

A sub-problem can be trivial: x < y or complex: search for a feasible flow

A sub-problem = a constraint

6

Constraints

Predefined constraints: arithmetic (x < y, x = y +z, |x-y| > k, alldiff, cardinality, sequence …

Constraints given in extension by the list of allowed (or forbidden) combinations of values

user-defined constraints: any algorithm can be encapsulated Logical combination of constraints using OR, AND, NOT, XOR

operators. Sometimes called meta-constraints

7

Filtering

We are able to solve a sub-problem: a method is available

CP uses this method to remove values from domain that do not belong to a solution of this sub-problem: filtering or domain-reduction

E.g: x < y and D(x)=[10,20], D(y)=[5,15]=> D(x)=[10,14], D(y)=[11,15]

8

Filtering

A filtering algorithm is associated with each constraint (sub-problem).

Can be simple (x < y) or complex (alldiff) Theoretical basics: arc consistency, remove all the

values that do not belong to a solution of the underlined sub-problem.

9

Propagation

Domain Reduction due to one constraint can lead to new domain reduction of other variables

When a domain is modified all the constraints involving this variable are studied and so on ...

10

Propagation

D(x)=D(y)={0,1}, D(z)={0,1,2}, D(u)={3,4,5}, D(v)={4,5,6}

Alldiff({x,y,z}), u=z+3, v>u, |z-v|<6

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Propagation

D(x)=D(y)={0,1}, D(z)={0,1,2}, D(u)={3,4,5}, D(v)={4,5,6}

Alldiff({x,y,z}), u=z+3, v>u, |z-v|<6• Alldiff({x,y,z}): FA=> z=2

12

Propagation

D(x)=D(y)={0,1}, D(z)={0,1,2}, D(u)={3,4,5}, D(v)={4,5,6}

Alldiff({x,y,z}), u=z+3, v>u, |z-v|<6• Alldiff({x,y,z}): FA=> z=2 u=z+3: FA=> u=5

13

Propagation

D(x)=D(y)={0,1}, D(z)={0,1,2}, D(u)={3,4,5}, D(v)={4,5,6}

Alldiff({x,y,z}), u=z+3, v>u, |z-v|<6• Alldiff({x,y,z}): FA=> z=2 u=z+3: FA=> u=5 v>u: FA=> v=6

14

Propagation

D(x)=D(y)={0,1}, D(z)={0,1,2}, D(u)={3,4,5}, D(v)={4,5,6}

Alldiff({x,y,z}), u=z+3, v>u, |z-v|<6• Alldiff({x,y,z}): FA=> z=2 u=z+3: FA=> u=5 v>u: FA=> v=6 |z-v|<6: FA => y=1

15

Propagation

D(x)=D(y)={0,1}, D(z)={0,1,2}, D(u)={3,4,5}, D(v)={4,5,6}

Alldiff({x,y,z}), u=z+3, v>u, |z-v|<6• Alldiff({x,y,z}): FA=> z=2 u=z+3: FA=> u=5 v>u: FA=> v=6 |z-v|<6: FA => y=1 Alldiff({x,y,z}): FA=> x=0 Solution: x=0, y=1, z=2, u=5, v=6

16

Why Propagation?

Idea: problem = conjunction of easy sub-problems. Sub-problems: local point of view. Problem: global

point of view. Propagation tries to obtain a global point of view from independent local point of view

The conjunction is stronger that the union of independent resolution

17

Search

Backtrack algorithm with strategies:try to successively assign variables with values. If a dead-end occurs then backtrack and try another value for the variable

Strategy: define which variable and which value will be chosen.

After each domain reduction (I.e assignement include) filtering and propagation are triggered

18

Constraint Programming

3 notions:- constraint network: variables, domains constraints + filtering (domain reduction)- propagation- search procedure (assignments + backtrack)

The structure of every constraint is exploited

19

Plan


20

CP vs other techniques

CP vs greedy algorithms CP vs local search CP vs inference engines CP vs MIP

21

CP vs greedy algorithm

Greedy algorithm = a search strategy which guarantees that there is no backtrack

CP is a generalization: your strategy can be non necessarily greedy. Solver manages for you the “errors” of your strategy

Some problems are solved with few backtracks

22

CP vs local search

Local search: it is hard to respect hard constraints. With CP no problem

Combination of CP and local search is used quite often to solve problems: find a first solution with CP then re-optimize it iteratively.

23

CP vs Inference engines

(A implies B) represented by (NOT A OR B) (x < 5 implies z > 4) Propagation from Left to Right: if x < 5 then z > 4 Propagation from Right to Left: if z < 5 then

necessarily x >4

24

CP vs MIPMIP approach CP approach

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CP vs MIP

Relax the problem: floats instead of integers

1) Use the Simplex algorithm (“polynomial”)

2) Set float to integer value. Go to 1) and backtrack if necessary

MIP approach CP approach

26

CP vs MIP




Identify sub-problems that are easy (called constraints)

1) Use specific algorithm for solving these sub-problems and for performing domain-reduction

2) Instantiate variable. Go to 1) and backtrack if necessary


27

CP vs MIP




Global point of view on a relaxation of the problem

Identify sub-problems that are easy (called constraints)

1) Use specific algorithm for solving these sub-problems and for performing domain-reduction

2) Instantiate variable. Go to 1) and backtrack if necessary

Local point of view on sub-problems. “Global” point of view by propagation of domain reductions


28

CP vs MIP

In CP constraints can be non-linear Structure of the problem is used in CP Semantic of the constraints are used: much easier to

add a global constraint with a filtering algorithm than defining a new cut

First solution given by CP is generally good

29

Plan


30

Arc consistency

All the values which do not belong to any solution of the constraint are deleted.

Example: Alldiff({x,y,z}) with D(x)=D(y)={0,1}, D(z)={0,1,2}the two variables x and y take the values 0 and 1, thus z cannot take these values.FA by AC => 0 and 1 are removed from D(z)

31

Alldiff constraint

x1

x2

x3

x4

x5

x6

1

2

3

4

5

6

7

The value graph:

D(x1)={1,2}D(x2)={2,3}D(x3)={1,3}D(x4)={3,4}D(x5)={2,4,5,6}D(x6)={5,6,7}

32

Value network

x1

x2

x3

x4

x5

x6

1

2

3

4

5

6

7

t s

(6,6)

(0,1)

(0,1)

(1,1)

Default orientation

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A feasible flow

x1

x2

x3

x4

x5

x6

1

2

3

4

5

6

7

t s

(6,6)

(0,1)

(0,1)

(1,1)

Default orientation

34

Residual graph

x1

x2

x3

x4

x5

x6

1

2

3

4

5

6

7

s

orientation

35

Residual graph

x1

x2

x3

x4

x5

x6

1

2

3

4

5

6

7

s

orientation

36

Arc consistency

x1

x2

x3

x4

x5

x6

1

2

3

4

5

6

7

The value graph:

D(x1)={1,2}D(x2)={2,3}D(x3)={1,3}D(x4)={4}D(x5)={5,6}D(x6)={5,6,7}

37

Alldiff constraint

Compute a feasible flow Compute the strongly connected components Remove every arc of flow value 0 for which the ends

belong to two different components Linear algorithm achieving arc consistency Idem for global cardinality constraints work well due to (0,1) arcs

38

Plan


39

The problem

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




• n teams and n-1 weeks and n/2 periods• every two teams play each other exactly once• every team plays one game in each week• no team plays more than twice in the same period

40

The problem







• Problem 10teams of the MIPLIB (n=10 and the objective function is dummy)• MIP is not able to find a solution for n=14• CP finds a solution for n=10 in 0.06s, n=14 in 0.2, n=40 in 6h

41

The problem







For 40 teams: 800 variables with 39 possible values foreach variable.

42

CP model: variables






For each slot: 2 variables represent the teams and 1 variable represents the match are defined

1 vs 6

T33a variable (T33a=6)T33h variable (T33h=1)

M33 variable (M33=12) Mij=1 <=> 0 vs 1 or 1 vs 0Mij=12 <=> 1 vs 6 or 6 vs1

43

CP model: T variables


Period 1 T11h vsT11a

T12h vsT12a

T13h vsT13a

T14h vsT14a

T15h vsT15a

T16h vsT16a

T17h vsT17a


T22h vsT22a

T23h vsT23a

T24h vsT24a

T25h vsT25a

T26h vsT26a

T27h vsT27a


T32h vsT32a

T33h vsT33a

T34h vsT34a

T35h vsT35a

T36h vsT36a

T37h vsT37a


T42h vsT42a

T43h vsT43a

T44h vsT44a

T45h vsT45a

T46h vsT46a

T47h vsT47a

D(Tija)=[1,n-1]D(Tijh)=[0,n-2]

Tijh < Tija

44

CP model: M variables


Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47

D(Mij)=[1,n(n-1)/2]

45

CP model: constraints


Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47

• n teams and n-1 weeks and n/2 periodsn teams and n-1 weeks and n/2 periods• every two teams play each other exactly once• every team plays one game in each weekevery team plays one game in each week• no team plays more than twice in the same periodno team plays more than twice in the same period

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Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47

• n teams and n-1 weeks and n/2 periodsn teams and n-1 weeks and n/2 periods• every two teams play each other exactly once• every team plays one game in each weekevery team plays one game in each week• no team plays more than twice in the same periodno team plays more than twice in the same period

Alldiff constraints defined on M variables

47




T12h vsT12a

T13h vsT13a

T14h vsT14a

T15h vsT15a

T16h vsT16a

T17h vsT17a


T22h vsT22a

T23h vsT23a

T24h vsT24a

T25h vsT25a

T26h vsT26a

T27h vsT27a


T32h vsT32a

T33h vsT33a

T34h vsT34a

T35h vsT35a

T36h vsT36a

T37h vsT37a


T42h vsT42a

T43h vsT43a

T44h vsT44a

T45h vsT45a

T46h vsT46a

T47h vsT47a


48




T12h vsT12a

T13h vsT13a

T14h vsT14a

T15h vsT15a

T16h vsT16a

T17h vsT17a


T22h vsT22a

T23h vsT23a

T24h vsT24a

T25h vsT25a

T26h vsT26a

T27h vsT27a


T32h vsT32a

T33h vsT33a

T34h vsT34a

T35h vsT35a

T36h vsT36a

T37h vsT37a


T42h vsT42a

T43h vsT43a

T44h vsT44a

T45h vsT45a

T46h vsT46a

T47h vsT47a


For each week w:Alldiff constraint definedon {Tpwh, p=1..4} U {Tpwa, p=1..4}

49




T12h vsT12a

T13h vsT13a

T14h vsT14a

T15h vsT15a

T16h vsT16a

T17h vsT17a


T22h vsT22a

T23h vsT23a

T24h vsT24a

T25h vsT25a

T26h vsT26a

T27h vsT27a


T32h vsT32a

T33h vsT33a

T34h vsT34a

T35h vsT35a

T36h vsT36a

T37h vsT37a


T42h vsT42a

T43h vsT43a

T44h vsT44a

T45h vsT45a

T46h vsT46a

T47h vsT47a


50




T12h vsT12a

T13h vsT13a

T14h vsT14a

T15h vsT15a

T16h vsT16a

T17h vsT17a


T22h vsT22a

T23h vsT23a

T24h vsT24a

T25h vsT25a

T26h vsT26a

T27h vsT27a


T32h vsT32a

T33h vsT33a

T34h vsT34a

T35h vsT35a

T36h vsT36a

T37h vsT37a


T42h vsT42a

T43h vsT43a

T44h vsT44a

T45h vsT45a

T46h vsT46a

T47h vsT47a


For each period p:Global cardinality constraint defined on{Tpwh, w=1..7} U {Tpwa, w=1..7}every team t is taken at most 2

51


For each slot the two T variables and the M variable must be linked together; example:M12 = game T12h vs T12a

52



For each slot we add Cij a ternary constraint defined on the two T variables and the M variable; example:C12 defined on {T12h,T12a,M12}

53




Cij are defined by the list of allowed tuples: for n=4: {(0,1,1),(0,2,2),(0,3,3),(1,2,4),(1,3,5),(2,3,6)}(1,2,4) means game 1 vs 2 is the game number 4

54




Cij are defined by the list of allowed tuples: for n=4: {(0,1,1),(0,2,2),(0,3,3),(1,2,4),(1,3,5),(2,3,6)}(1,2,4) means game 1 vs 2 is the game number 4

All these constraints have the same list of allowed tuples Efficient arc consistency algorithm for this kind of constraint is known

55

First model

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Dummy

Period 1 0 vs 1 0 vs 2 4 vs 7 3 vs 6 3 vs 7 1 vs 5 2 vs 4 . vs .




Introduction of a dummy column

56

First model







We can prove that:• each team occurs exactly twice for each period

57

First model


Period 1 0 vs 1 0 vs 2 4 vs 7 3 vs 6 3 vs 7 1 vs 5 2 vs 4 5 vs 6






58

First model








59

First model







We can prove that:• each team occurs exactly twice for each period• each team occurs exactly once in the dummy column

60

First model







• The problem is exactly the same• The solver is helped by such constraint. It can deduce someinconsistencies more quickly

61

First model: strategies

Break symmetries: 0 vs w appears in week w

62

First model: strategies

Break symmetries: 0 vs w appears in week w Teams are instantiated:

- the most instantiated team is chosen- the slots that has the less remaining possibilities (Tijh or Tija is minimal) is instantiated with that team

63

First model: results

# teams # fails Time (in s)

4 2 0.01

6 12 0.03

8 32 0.08

10 417 0.8

12 41 0.214 3,514 9.2

16 1,112 4.2

18 8,756 36

20 72,095 338

22 6,172,672 10h

24 6,391,470 12h

MIPLIB

MIP solver limit

64

Second model

Break symmetry: 0 vs 1 is the first game of the dummy column

65

Second model


1) Find a round-robin. Define all the games for each column (except for the dummy)- Alldiff constraint on M is satisfied- Alldiff constraint for each week is satisfied

66

Second model


1) Find a round-robin. Define all the games for each column (except for the dummy)- Alldiff constraint on M is satisfied- Alldiff constraint for each week is satisfied

2) set the games in order to satisfy constraints on periods. If no solution go to 1)

67

Second model: strategy


Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47

M variables are instantiated

68



Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47


69



Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47


70



Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47


71



Period 1 M11 M12 M13 M14 M15 M16 M17

Period 2 M21 M22 M23 M24 M25 M26 M27

Period 3 M31 M32 M33 M34 M35 M36 M37

Period 4 M41 M42 M43 M44 M45 M46 M47


72

Second model: results

# teams # fails Time (in s)

8 10 0.01

10 24 0.06

12 58 0.2

14 21 0.2

16 182 0.618 263 0.9

20 226 1.2

24 2702 10.5

26 5,683 26.4

30 11,895 138

40 2,834,754 6h

MIPLIB

MIP limit

First model limit

73

Assume P ≠ NP

Ok, we cannot avoid an exponential behavior For some instances, an NP Complete Problem will

required an exponential time to be solved So, our only hope is to shift the exponential such

that the problem is solvable for a size and a time that are acceptable

74

Shifting the exponential

0100200300400500600700800900

0 10 20 30 40 50

pb

75

Shifting the exponential

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50 60 70 80

pb

76

Why does CP perform well?

Pure discrete problem. You can give any number to the teams

This is a feasibility problem (no objective function). No arithmetic symbol: +, -, = is used A global point of view on the global cardinality

constraints (i.e. group these constraints into only one) does not help

77

Plan


78

Strength of CP

Very flexible (easy to take into account new constraints)

The system is open: you can define you own constraints, your own search mechanism.

CP allows the use of sophisticated strategies, you can use the knowledge of the domain of application.

You just have to respect a protocol given by a solver. The solver manages the propagation and provides you with a lot of predefined things

79

Strength of CP

CP is exact: no solution is lost even for float variables

Any existing algorithm can be integrated in CP as a filtering algorithm of a constraints

Concepts are simple A first model can be defined and tested quickly For optimization problems: the first solution is a good

one Easy to introduce your new “idea” in the system Cooperation is easy thanks to constraints

80

Plan


81

Weakness of CP

Must be improved for optimization problems: spend too much time in proving sub-optimality

First step: integration of cost in the constraints Sometimes lack of global point of view Dark zones: press Enter key then ? Relaxation is not good for CP. We learn relaxation at

school!

82

Plan


83

New research areas New point of view: CP is based on filtering algorithm, i.e. : Given a

property P defining a necessary condition for an element to be in a solutionFind as quickly as possible ALL elements that do not satisfy P

Ex: alldiff constraint and matching, cardinality constraint and flows etc…

Close to sensitivity analysis, but also different (for instance we only have monotonic modifications).

84

New research areas

Consider a Minimization problem, and OBJ the objective. Suppose that we found a “solution” with OBJ=25.

We will reject any “solution” with OBJ > 24. So if x=a leads to an OBJ > 24 then value a must be removed from D(x).

If we have a lower bound of OBJ then we can use it:if lb(OBJ,x=a) > 24 then remove a from D(x)

Problem: literature mainly gives upper bound for minimization problems and lower bound for maximization problems.

Not always easy to get lb of good quality

85

New research areas

The very same algorithm is called thousand times (million sometimes)

The incremental aspect of the algorithm becomes really important.

86

Plan


87

Recent advances in CP at ILOG

Let’s start with a real world example that involved a lot of people in ILOG:

The ROCOCO project From this project we learnt a lot of things

88

The ROCOCO Problem (1)

Routing of Communications Mono-routing: each demand from a point p to a point q

must follow a unique path Dimensioning of Links

The capacity of each link must exceed the sums of the demands going through the link

Additional Constraints Depend on the customer for whom the network is

designed

89


Data:• Customer traffic

demands• Possible links,

capacities and costs

S1 S2

S3

S4

S1S2

S3

S4

Result: Minimal cost

network able to simultaneously respond to all the demands

Route for each demand

27Kb/s

115Kb/s

Rented capacity 256Kb/s

90


Cost minimization principle Traffic demands share link capacities

S1 S2

S3

512Kb/s128Kb/s

115Kb/s

256Kb/s

91

The Problem (4)

Demands share links

demandsij capacityij

Technological constraints

256Kb/s

64Kb/s64Kb/s64Kb/s64Kb/s

128Kb/s

128Kb/s

64Kb/s64Kb/s

128Kb/s

92

The Problem (5)

Side constraints Quality of service Reuse of existing equipment (limit on the number of ports,

maximal traffic at a node)

Commercial and legal constraints Possible future network evolution Network management (e.g., traffic concentration)


93

Optional Constraints

Security: some commodities to be secured cannot go through unsecured nodes and links

No line multiplication: at most one line per arc. Symmetric routing: demands from node p to node q and demands from node q

to node p are routed on symmetric paths. Number of bounds (hops): the number of arcs of the path used to route a

given demand is limited. Number of ports: the number of links entering into or leaving from a node is

limited.

Maximal traffic: the total traffic managed by a given node is limited.


94

Numerical Characteristics (1)A10

05000

100001500020000250003000035000

0 100 200 300 400 500 600

Capacity

Co

st

B10

0

1000

2000

3000

4000

0 500 1000 1500 2000 2500

Capacity

Co

st

C10

0

500

1000

1500

2000

2500

0 10000 20000 30000 40000

Capacity

Co

st

B10-MULT

0

5000

10000

15000

20000

0 2000 4000 6000 8000 10000 12000

Capacity

Co

st

95

We worked a lot!

Average deviation from best-known solutions

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

A04 A06 A08 A10 B10 B12 B16 B25 C10 C12 C16 C25

CP-LS

CP-LNS02

CP-LNS03

CP-LNS04

CP-LNS04//

CP-LNS04//-8P

CP-LNS//-MOD1

CP-LNS//-MOD2

CP//

MIP

COLGEN

BPC02

96

The key idea: path variable!

97

General considerations

When solving a problem in CP: Potential performance gain:

data structure optimization (code): x 10 search strategies: x 1 000 model : x 1 000 000

Repartition of effort for ROCOCO data structure optimization (code): 65 % search strategies: 25 % model: 10 %

98

General considerations

When solving a problem in CP: Potential performance gain:

data structure optimization (code): x 10 search strategies: x 1 000 model : x 1 000 000

Repartition of effort for ROCOCO data structure optimization (code): 65 % search strategies: 25 % model: 10 %

Objective of CP Optimizer data structure optimization (code): 0 % search strategies: 10 % model: 90 %

99

Our current focus is on simplifying the solving process for the user

The user can then concentrate on modeling Why?

Huge ROI: a good model may lead to x1000000 improvement

The goal of CP is to be close to the natural description of the problem

Usability of our product will be improved

Focus CP at ILOG

100

Our current focus is on simplifying the solving process for the user

The user can then concentrate on modeling How?

Introduce a built-in intelligent search strategy Control the strategy only through higher-level

mechanisms Increase speed of the search

Increase inference strength (constraint propagation) Increase efficiency of the engine

Focus CP at ILOG

101

ILOG Solver is a mature product which has been on the market for fifteen years

We are embarking on a new product with a redesigned engine which we call ILOG CP

ILOG CP

102

Conclusion

CP is a general technique: can encapsulate a lot of work

CP is an efficient method for solving some combinatorial problems: small or large

Filtering algorithms are quite important for non binary constraints

CP allows the use of sophisticated strategies If you want to use CP: think CP (avoid Boolean (0-1)

variables). CP allows the use of symbolic representation

103

More information

Go to www.ilog.com Go to my webpage:

www.constraint-programming.com/people/reginor google my name: Jean Charles Regin

http://www.ilog.com/

http://www.constraint-programming.com/people/regin

general principles of constraint programming

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