Constraints and Search

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Constraints and Search
Logic AR Summer School, 2002

• Toby WalshCork Constraint Computation Centre
  (4C)
• tw_at_4c.ucc.ie

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Constraint satisfaction

• Constraint satisfaction problem (CSP) is a triple
  ltV,D,Cgt where
• V is set of variables
• Each X in V has set of values, D_X
• Usually assume finite domain
• true,false, red,blue,green, 0,10,
• C is set of constraints
• Goal find assignment of values to variables to
  satisfy all the constraints

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Constraint solver

• Tree search
• Assign value to variable
• Deduce values that must be removed from
  future/unassigned variables
• Constraint propagation
• If any future variable has no values, backtrack
  else repeat
• Number of choices
• Variable to assign next, value to assign
• Some important refinements like nogood learning,
  non-chronological backtracking,

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Constraint propagation

• Enfrocing arc-consistency (AC)
• A binary constraint r(X1,X2) is AC iff
• for every value for X1, there is a consistent
  value (often called support) for X2 and vice
  versa
• E.g. With 0/1 domains and the constraint X1 /
  X2
• Value 0 for X1 is supported by value 1 for X2
• Value 1 for X1 is supported by value 0 for X2
• A problem is AC iff every constraint is AC

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Tree search

• Backtracking (BT)
• Forward checking (FC)
• Maintaining arc-consistency (MAC)
• Limited discrepany search (LDS)
• Non-chronological backtracking learning

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Modelling case study orthogonal Latin squares

• Or constraint programming isnt purely
  declarative!

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Modelling decisions

• Many different ways to model even simple problems
• Its not pure declarative programming!
• 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
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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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2ns case study Langfords problem
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Langfords problem

• Prob024 _at_ 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

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Langfords problem

• L(k,n) problem
• To find a sequence of kn 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)

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Langfords problem

• 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 0ltnlt8, 10ltnlt17, 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

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Basic model

• What are the variables?

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Basic model

• 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

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Basic model

• What are the constraints?
• Xij in 1,nk
• Xij1 iXij
• Alldifferent(X11,..Xn1,X12,..Xn2,..,X1k,..Xnk)

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Recipe

• Create a basic model
• Decide on the variables
• Introduce auxiliary variables
• For messy/loose constraints
• Consider dual, combined or 0/1 models
• Break symmetry
• Add implied constraints
• Customize solver
• Variable, value ordering

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Break symmetry

• Does the problem have any symmetry?

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Break symmetry

• Does the problem have any symmetry?
• Of course, we can invert any sequence!

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Break symmetry

• How do we break this symmetry?

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Break symmetry

• How do we break this symmetry?
• Many possible ways
• For example, for L(3,9)
• Either X92 lt 14 (2nd occurrence of 9 is in 1st
  half)
• Or X9214 and X82lt14 (2nd occurrence of 8 is in
  1st half)

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Recipe

• Create a basic model
• Decide on the variables
• Introduce auxiliary variables
• For messy/loose constraints
• Consider dual, combined or 0/1 models
• Break symmetry
• Add implied constraints
• Customize solver
• Variable, value ordering

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What about dual model?

• Can we take a dual view?

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What about dual model?

• Can we take a dual view?
• Of course we can, its a permutation!

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Dual model

• What are the (dual) variables?
• Variable for each position i
• What are the values?

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Dual model

• What are the (dual) 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

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Dual model

• What are the (dual) variables?
• Variable for each position i
• What are the values?
• Solution 1 use values from 1,nk with the
  value inj standing for the ith occurrence of j
• Now want to find a permutation of these numbers
  subject to the distance constraint

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Dual model

• What are the (dual) 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

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Global cardinality constraint

• 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)
• Regins algorithm enforces GAC on Gcc in O(n2.d)
• Regins papers are tough to follow but this seems
  to beat his algorithm for all-different!?

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Dual model

• What are the constraints?
• Gcc(D1,Dkn,k,k)
• Distance constraints?

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Dual model

• What are the constraints?
• Gcc(D1,Dkn,k,k)
• Distance constraints
• Dij then Dij1j

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Combined model

• Primal and dual variables
• Channelling to link them
• What do the channelling constraints look like?

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Combined model

• Primal and dual variables
• Channelling to link them
• Xijk implies Dki

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Solving choices?

• Which variables to assign?
• Xij or Di

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Solving choices?

• Which variables to assign?
• Xij or Di, doesnt seem to matter
• Which variable ordering heuristic?
• Fail First or Lex?

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Solving choices?

• Which variables to assign?
• Xij or Di, doesnt seem to matter
• Which variable ordering heuristic?
• Fail First very marginally better than Lex
• How to deal with the permutation constraint?
• GAC on the all-different
• AC on the channelling
• AC on the decomposition

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Solving choices?

• Which variables to assign?
• Xij or Di, doesnt seem to matter
• Which variable ordering heuristic?
• Fail First very marginally better than Lex
• How to deal with the permutation constraint?
• AC on the channelling is often best for time

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Conclusions

• Modelling is an art but there are patterns
• Develop basic model
• Decide on the variables and their values
• Use auxiliary variables to represent constraints
  compactly/efficiently
• Consider dual, combined and 0/1 models
• Break symmetry
• Add implied constraints
• Customize solver for your model