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# Constraint Programming: modelling - PowerPoint PPT Presentation

Constraint Programming: modelling. Toby Walsh NICTA and UNSW. Golomb rulers. Mark ticks on a ruler Distance between any two ticks (not just neighbouring ticks) is distinct Applications in radio-astronomy, cystallography, … http://www.csplib.org/prob/prob006. Golomb rulers. Simple solution

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### Constraint Programming: modelling

Toby Walsh

NICTA and UNSW

• Mark ticks on a ruler

• Distance between any two ticks (not just neighbouring ticks) is distinct

• Applications in radio-astronomy, cystallography, …

• http://www.csplib.org/prob/prob006

• Simple solution

• Exponentially long ruler

• Ticks at 0,1,3,7,15,31,63,…

• Goal is to find minimal length rulers

• turn optimization problem into sequence of satisfaction problems

Is there a ruler of length m?

Is there a ruler of length m-1?

….

• Known for up to 23 ticks

• Distributed internet project to find large rulers

0,1

0,1,3

0,1,4,6

0,1,4,9,11

0,1,4,10,12,17

0,1,4,10,18,23,25

Solutions grow as approximately O(n^2)

• Variable, Xi for each tick

• Value is position on ruler

• Naïve model with quaternary constraints

• For all i>j,k>l>j |Xi-Xj| \= |Xk-Xl|

• Large number of quaternary constraints

• O(n^4) constraints

• Looseness of quaternary constraints

• Many values satisfy |Xi-Xj| \= |Xk-Xl|

• Limited pruning

• Introduce auxiliary variables for inter-tick distances

• Dij = |Xi-Xj|

• O(n^2) ternary constraints

• Post single large non-binary constraint

• alldifferent([D11,D12,…]).

• Tighter constraints and denser constraint graph

• Symmetry

• A ruler can always be reversed!

• Break this symmetry by adding constraint:

D12 < Dn-1,n

• Also break symmetry on Xi

X1 < X2 < … Xn

• Such tricks important in many problems

• Don’t change set of solutions

• But may reduce search significantly

E.g. D12 < D13, D23 < D24, …

E.g. D1k at least sum of first k integers

• Pure declarative specifications are not enough!

• Labeling strategies often very important

• Smallest domain often good idea

• Focuses on “hardest” part of problem

• Best strategy for Golomb ruler is instantiate variables in strict order

• Heuristics like fail-first (smallest domain) not effective on this problem!

• Circular (or modular) Golomb rulers

• Inter-tick distance variables more central, removing rotational symmetry?

• 2-d Golomb rulers

All examples of “graceful” graphs

• Modelling decisions:

• Auxiliary variables

• Implied constraints

• Symmetry breaking constraints

• More to constraints than just declarative problem specifications!

### Case study 2: all interval series

• Prob007 at www.csplib.org

• Comes from musical composition

• Traced back to Alban Berg

• Extensively used by Ernst Krenek

Op.170 “Quaestio temporis”

• Take the 12 standard pitch classes

• c, c#, d, ..

• Represent them by numbers 0, .., 11

• Find a sequence so each occurs once

• Each difference occurs once

• Can generalize to any n (not just 12)

Find Sn, a permutation of [0,n)

such that |Sn+1-Sn| are all distinct

• Finding one solution is easy

• Can generalize to any n (not just 12)

Find Sn, a permutation of [0,n) such that |Sn+1-Sn| are all distinct

• Finding one solution is easy

[n,1,n-1,2,n-2,.., floor(n/2)+2,floor(n/2)-1,floor(n/2)+1,floor(n/2)]

Giving the differences [n-1,n-2,..,2,1]

Challenge is to find all solutions!

• Devise basic CSP model

• What are the variables? What are the constraints?

• Introduce auxiliary variables if needed

• Consider dual or combined models

• Break symmetry

• Introduce implied constraints

• What are the variables?

• What are the variables?

Si = j if the ith note is j

• What are the constraints?

• What are the variables?

Si = j if the ith note is j

• What are the constraints?

Si in [0,n)

All-different([S1,S2,… Sn])

Forall i<i’ |Si+1 - Si| =/ |Si’+1 - Si’|

Will this model be any good? If so, why?

If not, why not?

• Devise basic CSP model

• What are the variables? What are the constraints?

• Introduce auxiliary variables if needed

• Consider dual or combined models

• Break symmetry

• Introduce implied constraints

• Is it worth introducing any auxiliary variables?

• Are there any loose or messy constraints we could better (more compactly?) express via some auxiliary variables?

• Is it worth introducing any auxiliary variables?

• Yes, variables for the pairwise differences

Di = |Si+1 - Si|

• Now post single large all-different constraint

Di in [1,n-1]

All-different([D1,D2,…Dn-1])

• Devise basic CSP model

• What are the variables? What are the constraints?

• Introduce auxiliary variables if needed

• Consider dual or combined models

• Break symmetry

• Introduce implied constraints

• Does the problem have any symmetry?

• Does the problem have any symmetry?

• Yes, we can reverse any sequence

S1, S2, … Sn is an all-inverse series

Sn, …, S2, S1 is also

• How do we eliminate this symmetry?

• Does the problem have any symmetry?

• Yes, we can reverse any sequence

S1, S2, …, Sn is an all-inverse series

Sn, …, S2, S1 is also

• How do we eliminate this symmetry?

• As with Golomb ruler!

D1 < Dn-1

• Does the problem have any other symmetry?

• Does the problem have any other symmetry?

• Yes, we can invert the numbers in any sequence

0, n-1, 1, n-2, … map x onto n-1-x

n-1, 0, n-2, 1, …

• How do we eliminate this symmetry?

• Does the problem have any other symmetry?

• Yes, we can invert the numbers in any sequence

0, n-1, 1, n-2, … map x onto n-1-x

n-1, 0, n-2, 1, …

• How do we eliminate this symmetry?

S1 < S2

• Devise basic CSP model

• What are the variables? What are the constraints?

• Introduce auxiliary variables if needed

• Consider dual or combined models

• Break symmetry

• Introduce implied constraints

• Are there useful implied constraints to add?

• Are there useful implied constraints to add?

• Hmm, unlike Golomb ruler, we only have neighbouring differences

• So, no need to consider transitive closure

• Are there useful implied constraints to add?

• Hmm, unlike Golomb ruler, we are not optimizing

• So, no need to improve propagation for optimization variable

• Basic model is poor

• Refined model able to compute all solutions up to n=14 or so

• GAC on all-different constraints very beneficial

• As is enforcing GAC on Di = |Si+1-Si|

This becomes too expensive for large n

So use just bounds consistency (BC) for larger n

### Case study 3: orthogonal Latin squares

• Many different ways to model even simple problems

• Combining models can be effective

• Channel between models

• Symmetry breaking

• Implied (but logically) redundant

• Each colour appears once on each row

• Each colour appears once on each column

• Used in experimental design

• Six people

• Six one-week drug trials

• Find a pair of Latin squares

• Every cell has a different pair of elements

• Generalized form:

• Find a set of m Latin squares

• Each possible pair is orthogonal

2 1 4 3 3 4 1 2

3 4 1 2 4 3 2 1

4 3 2 1 2 1 4 3

11 22 33 44

23 14 41 32

34 43 12 21

42 31 24 13

Two 4 by 4 Latin squares

No pair is repeated

Orthogonal Latin squares

• Introduced by Euler in 1783

• Also called Graeco-Latin or Euler squares

• No orthogonal Latin square of order 2

• There are only 2 (non)-isomorphic Latin squares of order 2 and they are not orthogonal

• Euler conjectured in 1783 that there are no orthogonal Latin squares of order 4n+2

• Constructions exist for 4n and for 2n+1

• Took till 1900 to show conjecture for n=1

• Took till 1960 to show false for all n>1

• 6 by 6 problem also known as the 36 officer problem

“… Can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6 by 6 array such that no row or column duplicates a rank or a regiment?”

• Lam’s problem

• Existence of finite projective plane of order 10

• Equivalent to set of 9 mutually orthogonal Latin squares of order 10

• In 1989, this was shown not to be possible after 2000 hours on a Cray (and some major maths)

• Orthogonal Latin squares also used in experimental design

• Suitable for integer programming

• Xijkl = 1 if pair (i,j) is in row k column l, 0 otherwise

• Avoiding advice never to use more than 3 subscripts!

• Constraints

• Each row contains one number in each square

Sum_jl Xijkl = 1 Sum_il Xijkl = 1

• Each col contains one number in each square

Sum_jk Xijkl = 1 Sum_ik Xijkl = 1

• Every pair of numbers occurs exactly once

Sum_kl Xijkl = 1

• Every cell contains exactly one pair of numbers

Sum_ij Xijkl = 1

Is there any symmetry?

• Important for solving CSPs

• Especially for proofs of optimality?

• Orthogonal Latin square has lots of symmetry

• Permute the rows

• Permute the cols

• Permute the numbers 1 to n in each square

• How can we eliminate such symmetry?

• Fix first row

11 22 33 …

• Fix first column

11

23

32

..

• Eliminates all symmetry?

• Exploit large finite domains possible in CSPs

• Reduce number of variables

• O(n^4) -> ?

• Exploit non-binary constraints

• Problem states that squares contain pairs that are all different

• All-different is a non-binary constraint our solvers can reason with efficiently

• 2 sets of variables

• Skl = i if the 1st element in row k col l is i

• Tkl = j if the 2nd element in row k col l is j

• How do we specify all pairs are different?

• All distinct (k,l), (k’,l’)

if Skl = i and Tkl = j then Sk’l’=/ i or Tk’l’ =/ j

O(n^4) loose constraints, little constraint propagation!

What can we do?

• Introduce auxiliary variables

• Fewer constraints, O(n^2)

• Tightens constraint graph => more propagation

• Pkl = i*n + j if row k col l contains the pair i,j

• Constraints

• 2n all-different constraints on Skl, and on Tkl

• All-different constraint on Pkl

• Channelling constraint to link Pkl to Skl and Tkl

3n^2 variables

Domains of size n, n and n^2+n

O(n^2) constraints

Large and tight non-binary constraints

0/1 model

n^4 variables

Domains of size 2

O(n^4) constraints

Loose but linear constraints

Use IP solver!

CSP model v O/1 model

• Variables to assign

• Skl and Tkl, or Pkl?

• Variable and value ordering

• How to treat all-different constraint

• GAC using Regin’s algorithm O(n^4)

• AC using the binary decomposition

• Experience and small instances suggest:

• Assign the Skl and Tkl variables

• Choose variable to assign with Fail First (smallest domain) heuristic

• Break ties by alternating between Skl and Tkl

• Use GAC on all-different constraints for Skl and Tkl

• Use AC on binary decomposition of large all-different constraint on Pkl

### Case study 4: Langford’s problem

• Prob024 @ www.csplib.org

• Find a sequence of 8 numbers

• Each number [1,4] occurs twice

• Two occurrences of i are i numbers apart

• Unique solution

• 41312432

• L(k,n) problem

• To find a sequence of k*n numbers [1,n]

• Each of the k successive occrrences of i are i apart

• We just saw L(2,4)

• Due to the mathematician Dudley Langford

• Watched his son build a tower which solved L(2,3)

• L(2,3) and L(2,4) have unique solutions

• L(2,4n) and L(2,4n-1) have solutions

• L(2,4n-2) and L(2,4n-3) do not

• Computing all solutions of L(2,19) took 2.5 years!

• L(3,n)

• No solutions: 0<n<8, 10<n<17, 20, ..

• Solutions: 9,10,17,18,19, ..

A014552

Sequence: 0,0,1,1,0,0,26,150,0,0,17792,108144,0,0,39809640,326721800,

0,0,256814891280,2636337861200

• What are the variables?

• What are the variables?

Variable for each occurrence of a number

X11 is 1st occurrence of 1

X21 is 1st occurrence of 2

..

X12 is 2nd occurrence of 1

X22 is 2nd occurrence of 2

..

• Value is position in the sequence

• What are the constraints?

• Xij in [1,n*k]

• Xij+1 = i+Xij

• Alldifferent([X11,..Xn1,X12,..Xn2,..,X1k,..Xnk])

• Create a basic model

• Decide on the variables

• Introduce auxiliary variables

• For messy/loose constraints

• Consider dual, combined or 0/1 models

• Break symmetry

• Customize solver

• Variable, value ordering

• Does the problem have any symmetry?

• Does the problem have any symmetry?

• Of course, we can invert any sequence!

• How do we break this symmetry?

• How do we break this symmetry?

• Many possible ways

• For example, for L(3,9)

• Either X92 < 14 (2nd occurrence of 9 is in 1st half)

• Or X92=14 and X82<14 (2nd occurrence of 8 is in 1st half)

• Create a basic model

• Decide on the variables

• Introduce auxiliary variables

• For messy/loose constraints

• Consider dual, combined or 0/1 models

• Break symmetry

• Customize solver

• Variable, value ordering

• Can we take a dual view?

• Can we take a dual view?

• Of course we can, it’s a permutation!

• What are the variables?

• Variable for each position i

• What are the values?

• What are the variables?

• Variable for each position i

• What are the values?

• If use the number at that position, we cannot use an all-different constraint

• Each number occurs not once but k times

• What are the variables?

• Variable for each position i

• What are the values?

• Solution 1: use values from [1,n*k] with the value i*n+j standing for the ith occurrence of j

• Now want to find a permutation of these numbers subject to the distance constraint

• What are the variables?

• Variable for each position i

• What are the values?

• Solution 2: use as values the numbers [1,n]

• Each number occurs exactly k times

• Fortunately, there is a generalization of all-different called the global cardinality constraint (gcc) for this

• Gcc([X1,..Xn],l,u) enforces values used by Xi to occur between l and u times

• All-different([X1,..Xn]) = Gcc([X1,..Xn],1,1)

• Regin’s algorithm enforces GAC on Gcc in O(n^2.d)

• Regin’s papers are tough to follow but this seems to beat his algorithm for all-different!?

• What are the constraints?

• Gcc([D1,…Dk*n],k,k)

• Distance constraints?

• What are the constraints?

• Gcc([D1,…Dk*n],k,k)

• Distance constraints:

• Di=j then Di+j+1=j

• Primal and dual variables

• What do the channelling constraints look like?

• Primal and dual variables

• Xij=k implies Dk=i

• Which variables to assign?

• Xij or Di

• Which variables to assign?

• Xij or Di, doesn’t seem to matter

• Which variable ordering heuristic?

• Fail First or Lex?

• Which variables to assign?

• Xij or Di, doesn’t seem to matter

• Which variable ordering heuristic?

• Fail First very marginally better than Lex

• Create a basic model

• Decide on the variables

• Introduce auxiliary variables

• For messy/loose constraints

• Consider dual, combined or 0/1 models

• Break symmetry