Case study 3: orthogonal Latin squares Modelled by Barbara Smith.

Case study 3: orthogonal Case study 3: orthogonal Latin squaresLatin squares

Modelled by Barbara SmithModelled by Barbara Smith

Modelling decisionsModelling decisions

Many different ways to model even simple Many different ways to model even simple problemsproblems

Combining models can be effectiveCombining models can be effective Channel between modelsChannel between models

Need additional constraintsNeed additional constraints Symmetry breakingSymmetry breaking Implied (but logically) redundantImplied (but logically) redundant

Orthogonal Latin squaresOrthogonal Latin squares

Find a pair of Latin Find a pair of Latin squaressquares Every cell has a different Every cell has a different

pair of elementspair of elements

Generalized form:Generalized form: Find a set of m Latin Find a set of m Latin

squaressquares Each possible pair is Each possible pair is

orthogonalorthogonal

Orthogonal Latin squaresOrthogonal Latin squares

1 2 3 4 1 2 3 41 2 3 4 1 2 3 42 1 4 3 3 4 1 22 1 4 3 3 4 1 23 4 1 2 4 3 2 13 4 1 2 4 3 2 14 3 2 1 2 1 4 34 3 2 1 2 1 4 3 11 22 33 4411 22 33 44 23 14 41 3223 14 41 32 34 43 12 2134 43 12 21 42 31 24 1342 31 24 13

Two 4 by 4 Latin Two 4 by 4 Latin squaressquares

No pair is repeatedNo pair is repeated

History of (orthogonal) Latin History of (orthogonal) Latin squaressquares

Introduced by Euler in 1783Introduced by Euler in 1783 Also called Graeco-Latin or Euler squaresAlso called Graeco-Latin or Euler squares

No orthogonal Latin square of order 2No orthogonal Latin square of order 2 There are only 2 (non)-isomorphic Latin squares of There are only 2 (non)-isomorphic Latin squares of

order 2 and they are not orthogonalorder 2 and they are not orthogonal

History of (orthogonal) Latin History of (orthogonal) Latin squaressquares

Euler conjectured in 1783 that there are no orthogonal Euler conjectured in 1783 that there are no orthogonal Latin squares of order 4n+2Latin squares of order 4n+2 Constructions exist for 4n and for 2n+1Constructions exist for 4n and for 2n+1 Took till 1900 to show conjecture for n=1Took till 1900 to show conjecture for n=1 Took till 1960 to show false for all n>1Took till 1960 to show false for all n>1

6 by 6 problem also known as the 36 officer problem6 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?”

More backgroundMore background

Lam’s problemLam’s problem Existence of finite projective plane of order 10Existence of finite projective plane of order 10 Equivalent to set of 9 mutually orthogonal Latin squares of Equivalent to set of 9 mutually orthogonal Latin squares of

order 10order 10 In 1989, this was shown not to be possible after 2000 hours In 1989, this was shown not to be possible after 2000 hours

on a Cray (and some major maths)on a Cray (and some major maths) Orthogonal Latin squares are used in experimental Orthogonal Latin squares are used in experimental

designdesign To ensure no dependency between independent variablesTo ensure no dependency between independent variables

A simple 0/1 modelA simple 0/1 model

Suitable for integer programmingSuitable for integer programming Xijkl = 1 if pair (i,j) is in row k column l, 0 Xijkl = 1 if pair (i,j) is in row k column l, 0

otherwiseotherwise Avoiding advice never to use more than 3 Avoiding advice never to use more than 3

subscripts!subscripts! ConstraintsConstraints

Each row contains one number in each squareEach row contains one number in each squareSum_jl Xijkl = 1 Sum_il Xijkl = 1Sum_jl Xijkl = 1 Sum_il Xijkl = 1

Each col contains one number in each squareEach col contains one number in each squareSum_jk Xijkl = 1 Sum_ik Xijkl = 1Sum_jk Xijkl = 1 Sum_ik Xijkl = 1

A simple 0/1 modelA simple 0/1 model

Additional constraintsAdditional constraints Every pair of numbers occurs exactly onceEvery pair of numbers occurs exactly once

Sum_kl Xijkl = 1Sum_kl Xijkl = 1 Every cell contains exactly one pair of numbersEvery cell contains exactly one pair of numbers

Sum_ij Xijkl = 1Sum_ij Xijkl = 1

Is there any symmetry?Is there any symmetry?

Symmetry removalSymmetry removal

Important for solving CSPsImportant for solving CSPs Especially for proofs of optimality?Especially for proofs of optimality?

Orthogonal Latin square has lots of symmetryOrthogonal Latin square has lots of symmetry Permute the rowsPermute the rows Permute the colsPermute the cols Permute the numbers 1 to n in each squarePermute the numbers 1 to n in each square

How can we eliminate such symmetry?How can we eliminate such symmetry?

Symmetry removalSymmetry removal

Fix first row Fix first row 11 22 33 …11 22 33 …

Fix first columnFix first column111123233232....

Eliminates all this symmetry?Eliminates all this symmetry?

What about a CSP model?What about a CSP model?

Exploit large finite domains possible in CSPsExploit large finite domains possible in CSPs Reduce number of variablesReduce number of variables O(n^4) -> ?O(n^4) -> ?

Exploit non-binary constraintsExploit non-binary constraints Problem states that squares contain pairs that are Problem states that squares contain pairs that are

all differentall different All-different is a non-binary constraint our solvers All-different is a non-binary constraint our solvers

can reason with efficientlycan reason with efficiently

CSP modelCSP model

2 sets of variables2 sets of variables Skl = i if the 1st element in row k col l is iSkl = i if the 1st element in row k col l is i Tkl = j if the 2nd element in row k col l is jTkl = j if the 2nd element in row k col l is j

How do we specify all pairs are different?How do we specify all pairs are different? All distinct (k,l), (k’,l’)All distinct (k,l), (k’,l’) if Skl = i and Tkl = j then Sk’l’=/ i or Tk’l’ =/ jif Skl = i and Tkl = j then Sk’l’=/ i or Tk’l’ =/ j

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

What can we do?What can we do?

CSP modelCSP model

Introduce auxiliary variablesIntroduce auxiliary variables Fewer constraints, O(n^2)Fewer constraints, O(n^2) Tightens constraint graph => more propagationTightens constraint graph => more propagation Pkl = i*n + j if row k col l contains the pair i,jPkl = i*n + j if row k col l contains the pair i,j

ConstraintsConstraints 2n all-different constraints on Skl, and on Tkl2n all-different constraints on Skl, and on Tkl All-different constraint on PklAll-different constraint on Pkl Channelling constraint to link Pkl to Skl and TklChannelling constraint to link Pkl to Skl and Tkl

CSP model v O/1 modelCSP model v O/1 model

CSP modelCSP model 3n^2 variables3n^2 variables Domains of size n, n and Domains of size n, n and

n^2+nn^2+n O(n^2) constraintsO(n^2) constraints Large and tight non-Large and tight non-

binary constraintsbinary constraints

0/1 model0/1 model n^4 variablesn^4 variables Domains of size 2Domains of size 2 O(n^4) constraintsO(n^4) constraints Loose but linear Loose but linear

constraintsconstraints Use IP solver!Use IP solver!

Solving choices for CSP modelSolving choices for CSP model

Variables to assignVariables to assign Skl and Tkl, or Pkl?Skl and Tkl, or Pkl?

Variable and value orderingVariable and value ordering How to treat all-different constraintHow to treat all-different constraint

GAC using Regin’s algorithm O(n^4)GAC using Regin’s algorithm O(n^4) AC using the binary decompositionAC using the binary decomposition

Good choices for the CSP modelGood choices for the CSP model

Experience and small instances suggest:Experience and small instances suggest: Assign the Skl and Tkl variablesAssign the Skl and Tkl variables Choose variable to assign with Fail First (smallest Choose variable to assign with Fail First (smallest

domain) heuristicdomain) heuristic Break ties by alternating between Skl and TklBreak ties by alternating between Skl and Tkl

Use GAC on all-different constraints for Skl and Use GAC on all-different constraints for Skl and TklTkl

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

PerformancePerformance

nn 0-1 model0-1 model

Fails t/secFails t/sec

CSP model CSP model ACAC


CSP model CSP model GACGAC


44 4 0.114 0.11 2 0.182 0.18 2 0.382 0.38

55 1950 4.051950 4.05 295 1.39295 1.39 190 1.55190 1.55

66 ? ?? ? 640235 657640235 657 442059 773442059 773

7* 7* 20083 59.820083 59.8 91687 51.191687 51.1 57495 66.157495 66.1

Dual CSP modelDual CSP model

4 subscripts in 0/1 model are interchangeable4 subscripts in 0/1 model are interchangeable Suggests a dual modelSuggests a dual model DSij = k if the pair (i,j) occurs in row kDSij = k if the pair (i,j) occurs in row k DTij = l if the pair (i,j) occurs in the row lDTij = l if the pair (i,j) occurs in the row l DPij = k*n + l if the pair (i,j) occurs in row i col jDPij = k*n + l if the pair (i,j) occurs in row i col j

Dual constraints to the primal modelDual constraints to the primal model

Combined modelCombined model

Primal and dual model togetherPrimal and dual model together Channelling constraints to link themChannelling constraints to link them

But new search decisionsBut new search decisions Do we assign both primal and dual variables?Do we assign both primal and dual variables? How do handle dual constraints (AC, GAC …)?How do handle dual constraints (AC, GAC …)? ……

Other dualities Other dualities Any choice of 2 subscripts from 4Any choice of 2 subscripts from 4 Diminishing returnsDiminishing returns

Dual performanceDual performance

NN PrimalPrimal


Dual Dual all varsall vars


Dual Dual primal varsprimal vars


44 2 0.382 0.38 0 0.290 0.29 0 0.0460 0.046

55 190 1.55190 1.55 94 0.3794 0.37 67 0.3467 0.34

66 442059 773442059 773 33868 14633868 146 26936 78.526936 78.5

7*7* 57495 66.157495 66.1 978 3.8978 3.8 1529 5.51529 5.5

ConclusionsConclusions

Many ways to model even simple problemsMany ways to model even simple problems Introduce auxiliary variablesIntroduce auxiliary variables

Reduce number of constraints, improve Reduce number of constraints, improve propagationpropagation

Combining models often beneficialCombining models often beneficial Channelling constraints link modelsChannelling constraints link models

Need to deal with symmetryNeed to deal with symmetry Don’t always use GAC on all-different Don’t always use GAC on all-different

constraintsconstraints

General methodology?General methodology?

Choose a basic modelChoose a basic model Consider auxiliary variablesConsider auxiliary variables

To reduce number of To reduce number of constraints, improve constraints, improve propagationpropagation

Consider combined modelsConsider combined models Channel between viewsChannel between views

Break symmetriesBreak symmetries Add implied constraintsAdd implied constraints

To improve propagationTo improve propagation

Case study 4: template Case study 4: template design design

Again model due to Barbara SmithAgain model due to Barbara Smith

IntroductionIntroduction

Prob002 at Prob002 at www.csplib.orgwww.csplib.org Problem comes from a printing firmProblem comes from a printing firm

Cat food labels that need to be printed using Cat food labels that need to be printed using templatestemplates

Several designs (tuna, chicken, …) go on each Several designs (tuna, chicken, …) go on each templatetemplate

Different demand for each flavourDifferent demand for each flavour Aside: where did cats get the taste for tuna?Aside: where did cats get the taste for tuna?

Template design problemTemplate design problem

For Francesca’s benefitFor Francesca’s benefit

How else can I get a cat How else can I get a cat picture into my talk?picture into my talk?

Basic CSP modelBasic CSP model

Assume number of templates is fixedAssume number of templates is fixed VariablesVariables

Pij = number of slots on template i for design jPij = number of slots on template i for design j Ri = run length for template iRi = run length for template i

ConstraintsConstraints Sum_j Pij = s, number of slots on each templateSum_j Pij = s, number of slots on each template Sum_i Pij * Ri >= dj, total production equals Sum_i Pij * Ri >= dj, total production equals

demanddemand

Basic CSP modelBasic CSP model

Optimization problemOptimization problem Introduce variable to minimizeIntroduce variable to minimize Production = Sum_i RiProduction = Sum_i Ri Solved as sequence of decision problemsSolved as sequence of decision problems

Production < l1, Production < l2 …Production < l1, Production < l2 …

l1 set to minimum number of printings with 1 templatel1 set to minimum number of printings with 1 template

SymmetrySymmetry

Does the model have any symmetry?Does the model have any symmetry?

If so, how can we eliminate it?If so, how can we eliminate it?

SymmetrySymmetry

The templates are indistinguishable and can be The templates are indistinguishable and can be permutedpermuted Swap all designs on one template with all those on Swap all designs on one template with all those on

a second templatea second template Break this symmetry by distinguishing the Break this symmetry by distinguishing the

templatestemplates R1 <= R2 <= R3 …R1 <= R2 <= R3 …

SymmetrySymmetry

Designs j, k with the same demand are Designs j, k with the same demand are indistinguishableindistinguishable

We can break this symmetryWe can break this symmetry [P1j,P2j,P3j,…] <lex [P1k,P2k,P3k,…][P1j,P2j,P3j,…] <lex [P1k,P2k,P3k,…] Efficient GAC algorithm for lex ordering Efficient GAC algorithm for lex ordering

constraintconstraint

Sort of symmetry?Sort of symmetry?

Symmetries can be subtle to spot!Symmetries can be subtle to spot! Consider designs j and j’ with demand for j Consider designs j and j’ with demand for j

less than for j’less than for j’ Suppose we produce more of j than j’Suppose we produce more of j than j’ We could swap j and j’ and still have solutionWe could swap j and j’ and still have solution Prevent this with constraint on productionPrevent this with constraint on production

Sum_i Pij Ri <= Sum_i Pij’ RiSum_i Pij Ri <= Sum_i Pij’ Ri

Implied constraintsImplied constraints

We should always look for implied constraints We should always look for implied constraints we can add to modelwe can add to model Encourage constraint propagationEncourage constraint propagation


2 templates2 templates R1+R2 = ProductionR1+R2 = Production R1 <= R2R1 <= R2 Hence Hence

R1 <= Production/2R1 <= Production/2

R2 >= Production/2R2 >= Production/2


3 templates3 templates R1 + R2 + R3 = ProductionR1 + R2 + R3 = Production R1 <= R2 <= R3R1 <= R2 <= R3 Hence Hence



R3 >= Production/3R3 >= Production/3

……

Solving choicesSolving choices

Variable orderingVariable ordering As with Golomb ruler, assign variables to As with Golomb ruler, assign variables to

construct solution systematicallyconstruct solution systematically Assign all designs on one template before moving Assign all designs on one template before moving

on to a second templateon to a second template Encourages constraint propagation on runlength Encourages constraint propagation on runlength

constraintsconstraints

PerformancePerformance

Basic modelBasic model Difficult to solve 2 or 3 template problemsDifficult to solve 2 or 3 template problems

Full modelFull model Problem solved quicklyProblem solved quickly Can solve much larger problems than feasible with Can solve much larger problems than feasible with

the basic modelthe basic model Optimality can still be tough!Optimality can still be tough!

ConclusionsConclusions

Basic model often obviousBasic model often obvious To refine such a model we need:To refine such a model we need:

Consider dual/combined modelsConsider dual/combined models Symmetries eliminatedSymmetries eliminated Implied constraintsImplied constraints Variable ordering heuristicsVariable ordering heuristics

Hopefully you can start to see patterns in what Hopefully you can start to see patterns in what we do!we do!

Case study 3: orthogonal Latin squares Modelled by Barbara Smith.

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