Case study 3: orthogonal Latin squares

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Case study 3 orthogonal Latin squares

• Modelled by Barbara Smith

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Modelling decisions

• Many different ways to model even simple problems
• Combining models can be effective
• Channel between models
• Need additional constraints
• Symmetry breaking
• Implied (but logically) redundant

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Orthogonal Latin squares

• 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

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Orthogonal Latin squares

• 1 2 3 4 1 2 3 4
• 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

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History of (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

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History of (orthogonal) Latin squares

• Euler conjectured in 1783 that there are no
  orthogonal Latin squares of order 4n2
• Constructions exist for 4n and for 2n1
• Took till 1900 to show conjecture for n1
• Took till 1960 to show false for all ngt1
• 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?

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More background

• Lams 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 are used in experimental
  design
• To ensure no dependency between independent
  variables

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A simple 0/1 model

• 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

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A simple 0/1 model

• Additional constraints
• 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?

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Symmetry removal

• 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?

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Symmetry removal

• Fix first row
• 11 22 33
• Fix first column
• 11
• 23
• 32
• ..
• Eliminates all this symmetry?

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What about a CSP model?

• Exploit large finite domains possible in CSPs
• Reduce number of variables
• O(n4) -gt ?
• 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

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CSP model

• 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 Skl/ i or
  Tkl / j
• O(n4) loose constraints, little constraint
  propagation!
• What can we do?

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CSP model

• Introduce auxiliary variables
• Fewer constraints, O(n2)
• Tightens constraint graph gt more propagation
• Pkl in 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

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CSP model v O/1 model

• CSP model
• 3n2 variables
• Domains of size n, n and n2n
• O(n2) constraints
• Large and tight non-binary constraints

• 0/1 model
• n4 variables
• Domains of size 2
• O(n4) constraints
• Loose but linear constraints
• Use IP solver!

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Solving choices for CSP model

• Variables to assign
• Skl and Tkl, or Pkl?
• Variable and value ordering
• How to treat all-different constraint
• GAC using Regins algorithm O(n4)
• AC using the binary decomposition

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Good choices for the CSP model

• 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

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Performance
n 0-1 model Fails t/sec CSP model AC Fails t/sec CSP model GAC Fails t/sec
4 4 0.11 2 0.18 2 0.38
5 1950 4.05 295 1.39 190 1.55
6 ? ? 640235 657 442059 773
7 20083 59.8 91687 51.1 57495 66.1
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Dual CSP model

• 4 subscripts in 0/1 model are interchangeable
• Suggests a dual model
• DSij k if the pair (i,j) occurs in row k
• DTij l if the pair (i,j) occurs in the row l
• DPij kn l if the pair (i,j) occurs in row i
  col j
• Dual constraints to the primal model

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Combined model

• Primal and dual model together
• Channelling constraints to link them
• But new search decisions
• Do we assign both primal and dual variables?
• How do handle dual constraints (AC, GAC )?
• Other dualities
• Any choice of 2 subscripts from 4
• Diminishing returns

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Dual performance
N Primal Fails t/sec Dual all vars Fails t/sec Dual primal vars Fails t/sec
4 2 0.38 0 0.29 0 0.046
5 190 1.55 94 0.37 67 0.34
6 442059 773 33868 146 26936 78.5
7 57495 66.1 978 3.8 1529 5.5
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Conclusions

• Many ways to model even simple problems
• Introduce auxiliary variables
• Reduce number of constraints, improve propagation
• Combining models often beneficial
• Channelling constraints link models
• Need to deal with symmetry
• Dont always use GAC on all-different constraints

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General methodology?

• Choose a basic model
• Consider auxiliary variables
• To reduce number of constraints, improve
  propagation
• Consider combined models
• Channel between views
• Break symmetries
• Add implied constraints
• To improve propagation

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Case study 4 template design

• Again model due to Barbara Smith

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Introduction

• Prob002 at www.csplib.org
• Problem comes from a printing firm
• Cat food labels that need to be printed using
  templates
• Several designs (tuna, chicken, ) go on each
  template
• Different demand for each flavour
• Aside where did cats get the taste for tuna?

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Template design problem

• For Francescas benefit
• How else can I get a cat picture into my talk?

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Basic CSP model

• Assume number of templates is fixed
• Variables
• Pij number of slots on template i for design j
• Ri run length for template i
• Constraints
• Sum_j Pij s, number of slots on each template
• Sum_i Pij Ri gt dj, total production equals
  demand

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Basic CSP model

• Optimization problem
• Introduce variable to minimize
• Production Sum_i Ri
• Solved as sequence of decision problems
• Production lt l1, Production lt l2
• l1 set to minimum number of printings with 1
  template

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Symmetry

• Does the model have any symmetry?
• If so, how can we eliminate it?

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Symmetry

• The templates are indistinguishable and can be
  permuted
• Swap all designs on one template with all those
  on a second template
• Break this symmetry by distinguishing the
  templates
• R1 lt R2 lt R3

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Symmetry

• Designs j, k with the same demand are
  indistinguishable
• We can break this symmetry
• P1j,P2j,P3j, ltlex P1k,P2k,P3k,
• Efficient GAC algorithm for lex ordering
  constraint

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Sort of symmetry?

• Symmetries can be subtle to spot!
• Consider designs j and j with demand for j less
  than for j
• Suppose we produce more of j than j
• We could swap j and j and still have solution
• Prevent this with constraint on production
• Sum_i Pij Ri lt Sum_i Pij Ri

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Implied constraints

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

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Implied constraints

• 2 templates
• R1R2 Production
• R1 lt R2
• Hence
• R1 lt Production/2
• R2 gt Production/2

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Implied constraints

• 3 templates
• R1 R2 R3 Production
• R1 lt R2 lt R3
• Hence
• R1 lt Production/3
• R2 lt Production/2
• R3 gt Production/3

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Solving choices

• Variable ordering
• As with Golomb ruler, assign variables to
  construct solution systematically
• Assign all designs on one template before moving
  on to a second template
• Encourages constraint propagation on runlength
  constraints

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Performance

• Basic model
• Difficult to solve 2 or 3 template problems
• Full model
• Problem solved quickly
• Can solve much larger problems than feasible with
  the basic model
• Optimality can still be tough!

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Conclusions

• Basic model often obvious
• To refine such a model we need
• Consider dual/combined models
• Symmetries eliminated
• Implied constraints
• Variable ordering heuristics
• Hopefully you can start to see patterns in what
  we do!