Modelling for Constraint Programming Barbara Smith [email protected] CP 2010 Doctoral Programme.

Modelling for Constraint ProgrammingBarbara Smith

[email protected]

CP 2010 Doctoral Programme

Context• I will assume

– A well-defined problem that can be represented as a finite domain constraint satisfaction or optimization problem

• no uncertainty, preferences, etc. – A constraint solver providing:

• a systematic search algorithm • combined with constraint propagation• a set of pre-defined constraints

• I am not going to discuss– Automated modelling– Choice of search strategy– Global constraints– etc.

• There won’t be enough examples!


Solving CSPs

• Systematic search: given a CSP M = <X,D,C> – choose a variable xi that is not yet assigned

– create a choice point, i.e. a set of mutually exclusive & exhaustive choices, e.g. xi = a v. xi ≠ a

– try the first & backtrack to try the other if this fails

• Constraint propagation: – add xi = a or xi ≠ a to the set of constraints

– re-establish local consistency on each constraint • remove values from the domains of future variables that can no

longer be used because of this choice

– fail if any future variable has no values left


Representing a Problem• If a CSP M = <X,D,C> represents a problem P, then every

solution of M corresponds to a solution of P and every solution of P can be derived from at least one solution of M

• More than one solution of M can represent the same solution of P, if modelling introduces symmetry

• The variables and values of M represent entities in P• The constraints of M ensure the correspondence between

solutions of M and solutions of P• The aim is (usually) to find a model M that can be solved as

quickly as possible– NB shortest run-time might not mean least search


Interactions with Search Strategy • Whether M1 is better than M2 can depend on the search

algorithm and search strategy• I will assume the search algorithm is fixed and choice points

are always xi = a v. xi ≠ a

• But the search strategy is still important – choice of search variables– variable and value ordering heuristics

• E.g. Stable marriage, Graceful labelling of a graph• Is designing the search strategy part of modelling?

– I think it is, in practice – but I will not discuss it in this talk


Viewpoints• A viewpoint is a pair <X,D>, i.e. a set of variables and their

domains• Given a viewpoint, the constraints have to restrict the

solutions of M to solutions of P– So the constraints are (largely) decided by the viewpoint– Different viewpoints give very different models

• We can combine viewpoints - more later• Good rule of thumb: choose a viewpoint that allows the

constraints to be expressed easily and concisely– will propagate well, so problem can be solved efficiently


Example: Magic Square

• Arrange the numbers 1 to 9 in a 3 x 3 square so that each row, column and diagonal has the same sum

• V1 : a variable for each cell, domain is the numbers that can go in the cell

• V2 : a variable for each number, domain is the cells where that number can go

• Constraints on row, column & diagonal sums are easy to express in V1:– x1+x2+x3 = x4+x5+x6 = x1+x4+x7 = …

• but not in V2

4 3 8

9 5 1

2 7 6

x1 x2 x3

x4 x5 x6

x7 x8 x9


Constraints

• Given a viewpoint, the role of the constraints is:– To ensure that the solutions of the CSP match the solutions of

the problem

– To guide the search, i.e. to ensure that as far as possible, partial

solutions that will not lead to a solution fail immediately


Expressing the Constraints

• For efficient solving, we need to know:– the constraints provided by the constraint solver– the level of consistency enforced on each– the complexity of the constraint propagation algorithms– Not very declarative!

• There is often a trade-off between time spent on propagation and time saved on search – which choice is best often depends on the problem– E.g. sometimes it is not worthwhile to enforce GAC on an

allDifferent constraint• If the constraint is loose, i.e. there are many more values than variables


Auxiliary Variables• Often, the constraints can be expressed more

easily/propagate better if more variables are introduced• E.g. in the magic square problem, introduce a variable s

representing the sum of the rows & columns– x1+x2+x3 = x4+x5+x6 becomes x1+x2+x3 = s, x4+x5+x6 = s

– Better still, add s = 15 (in the 3 × 3 case)


Auxiliary Variables (II)• Example: car sequencing (Dincbas, Simonis and van

Hentenryck, ECAI 1988)


• 10 cars to be made on a production line, each requires some options

• Stations installing options have lower capacity than rest of line e.g. at most 1 car out of 2 for option 1

• Find a feasible production sequence

classes 1 2 3 4 5 6 Capacity

Option 1 1 0 0 0 1 1 1/2

Option 2 0 0 1 1 0 1 2/3

Option 3 1 0 0 0 1 0 1/3

Option 4 1 1 0 1 0 0 2/5

Option 5 0 0 1 0 0 0 1/5

No. of cars

1 1 2 2 2 2

Car Sequencing - Model• Variables : s1 , s2 , …, s10 • Value of si is the class of car in position i in the sequence• Constraints:

– Each class occurs the correct number of times– Option capacities are respected - ?

• Introduce variables oij :– oij = 1 iff the car in the i th slot in the sequence requires option j

• Option 1 capacity is one car in every two:– oi,1 + oi+1,1 ≤ 1 for 1 ≤ i < 10

• Relate the auxiliary variables to the si variables :– λjk = 1 if car class k requires option j – oij = λjsi , 1 ≤ i ≤ 10, 1 ≤ j ≤ 5


One Constraint is Better than Several (maybe)

• If there are several constraints all with the same scope, rewriting them as a single constraint will lead to more propagation…– if the same level of consistency is maintained on the

new constraint• … more propagation means shorter run-time

– if enforcing consistency on the new constraint can be done efficiently

• NB conjoining all the constraints in the problem into a single constraint is not usually a good idea!


Example: n-queens

• A variable for each row, x1 , x2 , …, xn

• Values represent the columns, 1 to n• The assignment (xi,c) means that the queen in row i is

in column c• Constraints for each pair of rows i, j:

– xi ≠xi

– xi − xi ≠ i − j

– xi − xi ≠ j − i


Propagating the Constraints

• A queen in row 5, column 3 conflicts with both remaining values for x3

• But the constraints are consistent– xi ≠xi thinks that (x3 ,1) can support

(x5 ,3) – xi − xi ≠ i − j thinks that (x3 ,3) can

support (x5 ,3)


×

• Enforcing AC on the conjunction (xi ≠xi ) ٨ (xi

− xi ≠ i − j ) ٨ (xi − xi ≠ j − i ) would remove 3 from the domain of x5

• If you can enforce GAC on the constraint


Implied Constraints

• Implied constraints are logical consequences of the set of existing constraints– So are logically redundant (sometimes called

redundant constraints)

• They do not change the set of solutions• Adding implied constraints can reduce the

search effort and run-time


Example: Car Sequencing• Existing constraints only say that the option capacities cannot

be exceeded • Suppose there are 30 cars and 12 require option 1 (capacity

1/2)• At least one car in slots 1 to 8 of the production sequence

must require option 1; otherwise 12 of cars 9 to 30 will require option 1, i.e. too many

• Cars 1 to 10 must include at least two option 1 cars, ... , and cars 1 to 28 must include at least 11 option 1 cars

• These are implied constraints


Useful Implied Constraints

• An implied constraint reduces search if:– at some point during search, a partial assignment will fail because of

the implied constraint– without the implied constraint, the search would continue– the partial assignment cannot lead to a solution

• the implied constraint forbids it, but does not change the set of solutions

• In car sequencing, partial assignments with option 1 under-used could be explored during search, without the implied constraints


Useless Implied Constraints

• The assignments forbidden by an implied constraint may never actually arise– depends on the search order

• e.g. in car sequencing, – at least one of cars 1 to 8 must require option 1 – any 8 consecutive cars must have one option 1 car– but if the sequence is built up from slot 1, only the implied

constraints on slots 1 to k can cause the search to backtrack• If we find a class of implied constraints, maybe only some

are useful– adding a lot of constraints that don’t reduce search will increase

the run-time


Implied Constraints v. Global Constraints

• Régin and Puget (CP97) developed a global constraint for sequence problems, including the car sequencing problem – “our filtering algorithm subsumes all the implied constraints” used

by Dincbas et al. • Implied constraints may only be useful because a suitable

global constraint does not (yet) exist • But many implied constraints are simple and quick to

propagate• Use a global constraint if there is one available and it is

cost-effective– but look for useful implied constraints as well


Symmetry in CSPs

• A symmetry transforms any solution into another– Sometimes symmetry is inherent in the problem

(e.g. chessboard symmetry in n-queens)– Sometimes it’s introduced in modelling

• Symmetry causes wasted search effort: after exploring choices that don’t lead to a solution, symmetrically equivalent choices may be explored


Symmetry between Values: Car sequencing

cars 1 2 3 4 5 6 7 8 9 10

option 1

1

0 0 0 0 0 1 1 1 1

option 2

0 0 1 1 1 1 0 0 1 1

option 3

1 0 0 0 0 0 1 1 0 0

option 4

1

1 0 0 1 1 0 0 0 0

option 5

0 0 1 1 0 0 0 0 0 0

classes 1 2 3 4 5 6

option 1 1 0 0 0 1 1

option 2 0 0 1 1 0 1

option 3 1 0 0 0 1 0

option 4 1 1 0 1 0 0

option 5 0 0 1 0 0 0

no. of cars

1 1 2 2 2 2

• A natural model has individual cars as the values • introduces symmetry

between cars requiring the same option

• The model instead has classes of car• needs constraints to

ensure the right number of cars in each class

• Using set variables is a similar and common technique to remove symmetry


Symmetry Breaking

• Often, not all the symmetry can be eliminated by remodelling

• Remaining symmetry should be reduced or eliminated:– dynamic symmetry breaking methods (SBDS,

SBDD, etc.)– symmetry-breaking constraints

• unlike implied constraints, they change the set of solutions

• can lead to further implied constraints


Changing Viewpoint

• We can improve a CSP model of a problem– express the constraints better– break the symmetry – add implied constraints

• But sometimes it’s better just to use a different model– i.e. a different viewpoint


Different Viewpoints

• Reformulate in a standard way, e.g.– non-binary to binary translations – dual viewpoint for permutation problems– Boolean to integer or set viewpoints– Reformulations are being developed for automated

modelling • Or find a new viewpoint by viewing the problem

from a different angle – the constraints may express different insights into

the problem


Permutation Problems

• A CSP is a permutation problem if: – it has the same number of values as variables– all variables have the same domain – each variable must be assigned a different value

• Any solution assigns a permutation of the values to the variables

• Other constraints determine which permutations are solutions

• There is a dual viewpoint in which the variables and values are swapped


Example: n-queens

• Standard model– a variable for each row, x1 , x2 , …, xn

– values represent the columns, 1 to n– xi = j means that the queen in row i is in column j

– n variables, n values, allDifferent(x1 , x2 , …, xn)

• Dual viewpoint– a variable for each column, d1 , d2 , …, dn ; values represent the

rows

• In this problem, both viewpoints give the same CSP


Example: Magic Square

• First viewpoint:– variables x1 , x2 , …, x9

– values represent the numbers 1 to 9– The assignment (xi ,j) means that the number

in square i is j• Dual viewpoint

– a variable for each number, d1 , d2 , …, d9 – values represent the squares

• Constraints are much easier to express in the first viewpoint (& there are far fewer of them)

x1 x2 x3

x4 x5 x6

x7 x8 x9


Different Perspectives: Example

• Constraint Modelling Challenge, IJCAI 05• “Minimizing the maximum number of open

stacks”• There were almost as many models of the

problem as participants in the Challenge


Which Viewpoint to Choose?

• Sometimes one viewpoint is clearly better, e.g. if we can’t express the constraints easily in one

• But different perspectives often allow different expression of the constraints and different implied constraints– can be hard to decide which is better

• We don’t need to choose one viewpoint – we can use two (or more) at once

• We need channelling constraints to link the variables


Combining Viewpoints: Permutation Problems

• Dual viewpoints of a permutation problem with variables x1, x2, …, xn and d1, d2, …, dn

• Combine them using the channelling constraints (xi = j) ≡ (dj = i )

• Also allows both sets of variables to be search variables– e.g. use a dynamic variable order e.g. variable with

smallest domain in either viewpoint – combines variable and value ordering: dual variable

with smallest domain corresponds to the value occurring in fewest domains (in the other viewpoint)


Combining Viewpoints: Integer & Set Variables

• In a nurse rostering problem, we can allocate shifts to nurses or nurses to shifts

• First viewpoint: – an integer variable nij for each nurse i and day j– its value is the shift that nurse i works on day j

• Second viewpoint: – a set variable Skj for each shift k and day j– its value is the set of nurses that work shift k on day j

• Channelling constraints: (nij = k) ≡ (i Skj) • Constraints on nurse availability are stated in the first

viewpoint; constraints on work requirements in the 2nd (e.g. no. of nurses required for each shift)


Search Variables – Permutation Problems

• We can use both sets of variables as search variables– e.g. use a dynamic variable order e.g. variable

with smallest domain in either viewpoint • combines variable and value ordering: dual variable

with smallest domain corresponds to the value occurring in fewest domains (in the other viewpoint)

Summary• Choose a viewpoint

• that allows the constraints to be expressed easily and concisely• Use auxiliary variables if they help to express the constraints • Implied constraints can detect infeasible subproblems earlier – Make sure they are useful• Look out for symmetry in the CSP– avoid it if possible by changing the model– eliminate it e.g. by adding constraints

• Consider alternative viewpoints– think of standard reformulations– think about the problem in different ways – consider combining viewpoints



Conclusion• Aim for a rich model

– multiple viewpoints– auxiliary variables– add constraints to improve propagation (implied

constraints, global constraints) & symmetry breaking constraints

• But check that your additions do reduce run-time with your chosen search strategy

• Understand the problem as well as you can– build that insight into the model– the better you can understand a problem, the better

you can solve itTHE END

Modelling for Constraint Programming Barbara Smith [email protected] CP 2010 Doctoral Programme.

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Transcript of Modelling for Constraint Programming Barbara Smith [email protected] CP 2010 Doctoral Programme.