10.3 Reformulation 10.4.1 The Lex-Leader Method

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10.3 Reformulation10.4.1 The Lex-Leader Method

• Shant Karakashian

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Outline

• Introduction Modeling Reformulation
• Examples
• Social Golfers Problem Set Variables and
  Difficulties
• n-integer permutation and Reformulation
• Reformulation Techniques
• Different Viewpoints
• Prestwich's Reformulation
• The Properties of Reformulation
• Adding Constraints Before Search
• Introduction to Lex-Leader Method
• Lex-Leader Method
• The idea
• Procedure
• Lex-Leader and Variable Ordering
• Limitation

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Introduction

• Modelling a problem has a substantial effect on
  the efficiently of the problem solving
• Different models of the same problem can have
  different symmetries
• A problem can be reformulated thus getting
  different symmetries.
• Modeling and reformulation are important for
  symmetry breaking
• Once a problem is reformulated, the remaining
  symmetries can be dealt with before or during
  search

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Social Golfers Problem

• Golfers play once a week in groups, such that no
  two people play in the same group in two
  different weeks
• 32 golfers
• 8 groups of 4 people each
• 10 weeks 
• The problem has 32!10!8!104!80 symmetries.
• Remodel the problem to get 32!10! symmetries
• B. Smith '04
• Adding a variable for each pair of players to
  indicate which week they played together

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Set Variables

• Given a number of indistinguishable variables
• Encode the variables as a set of variables
• In the social golfers' example
• Encode the groups playing within each week as
  sets (total of 8 sets per week)
• The size of each set is 4
• Every pair of sets constraint to have empty
  intersection
• Result lose 4! symmetries in each group
• Total reduction of symmetries 2480

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Difficulties with Set Variables

• There are theoretical and practical difficulties
  associated with set variables
• Representations of set variables by the solver
  can have dramatically different behaviors in
  propagation
• e.g. if the used representation is not suitable
  for the constraints associated with the
  variables, search can be dramatically increased. 
• In cases where set and integer variables are
  mixed 
• "Channeling" between  set and integer variables
  can be difficult, and lead to delaying the
  propagation until late in search. 
•  Reformulating is an art, not a science. 

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Example n-integer permutation

• Given n integers from 0 to n-1, find a
  permutation so that the differences between
  adjacent numbers are also a permutation of the
  numbers 1 to n-1.
• n11 
• Four obvious symmetries in the problem
• identity
• reversing the series
• negating each element by subtracting it from
  (n-1)
• reversing and negating

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Reformulation of n-integer permutation Gent et
al. '04

• Given a solution to the problem, cycle it about a
  pivot to generate another solution
• The difference between first and last terms must
  always duplicate a difference in the sequence, so
  this operation can be applied to any solution

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Reformulation of n-integer permutation (contd.)

• Reformulate the problem to solve a different
  problem
• Solutions of reformulated problem lead to
  solutions to the original problem
• Given n integers from 0 to n-1, find a
  permutation so that
• The differences between adjacent numbers and the
  two extreme numbers are also a permutation of the
  numbers 1 to n-1
• The sequence has to obey two constraints 
• The permutation starts 0,n-1,1
• The n differences between consecutive numbers
  contain all of 1,...n-1 with one
  difference occurring exactly twice

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Different Viewpoints

• Given a CSP with
• n variables
• single domain for all variables of size n
• all variables must take different values
• Then there are different 'Viewpoints" of the
  problem
• Find values for each variable
• Find variables for each value
• If there is symmetry
• Value symmetry in the first viewpoint is
  interchanged with variable symmetry in the second
  and vice versa

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Prestwich's Reformulation

• A significant advance in understanding how
  reformulation can be applied mechanically
• Prestwich showed that value symmetries can be
  eliminated automatically by a new encoding of
  CSPs to SAT
• Breaks all value symmetries of a special kind
• Called 'Dynamic Substitutability
• Is a variant of Freuder's value
  interchangeability
• Prestwich's encoding eliminates all dynamic
  substitutability
• Symmetry detection not necessary

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The Properties of the Reformulation

• Disadvantages
• Limited to certain types of value symmetry
• No guarantee that the reformulation would lead to
  improved search 
• Adventages
• From a black art to a science
• The concerns are on tradeoffs and implementation
• No need for magical insights

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Adding Constraints Before Search

• Symmetry breaking constraints when added in an ad
  hoc fashion
• Need to recognize the symmetry
• Variables are indistinguishable
• force them to be in non-decreasing order.
• Variables are indistinguishable and must have
  different values
• Force them to be in strictly increasing order
• Can loose solutions if incorrectly done

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Introduction to Lex-Leader Method

• A CSP has symmetry
• Eliminated by adding constraints to the original
  problem
• Possible to find a 'reduced form
• Crawford et al. '96 outlined the technique
  lex-leader
• For Variable symmetries
• Constructs symmetry-breaking ordering constraints

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The idea of Lex-Leader Method

• For a symmetry group
• For each equivalence class of solutions
• Predefine one to be the canonical solution
• Achieved by
• Adding constraints before search
• Satisfy only the canonical solutions
• To induce an ordering on full assignments
• Requires choosing a static variable ordering

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Lex-Leader Method

• Defined on variable symmetries
• For every permutation g converting a tuple into
  another tuple
• Choose the one that gives the value that is
  lexicographically least
• example
• variables A, B, C
• constraints all differnet
• Permutations A B C, A C B, B A C, B C A, C A B,
  C B A
• Solutions A1 B2 C3 and A1 B3 C2 are
  equivalent
• With constraint A B C lt A C B only 1 2 3 is a
  solution
• For 1 3 2 1 3 2 gt 1 2 3

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Lex-Leader Example

• Consider a 3x2 matrix in a context
• The rows and columns may be freely permuted
• The symmetries form the group S3 x S2
• Group order is 3!2! 12
• ABCDEF lex ABCDEF
• ABCDEF lex ACBDFE
• ABCDEF lex BCAEFD
• ABCDEF lex BACEDF
• ABCDEF lex CABFDE
• ABCDEF lex CBAFED
• ABCDEF lex DFEACB
• ABCDEF lex EFDBCA
• ABCDEF lex EDFBAC
• ABCDEF lex FDECAB
• ABCDEF lex FEDCBA
• ABCDEF lex DEFABC

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Lex-Leader and Variable Ordering

• Does not respect the variable and value ordering
  heuristics
• If the leftmost solution in the search tree is
  not canonical
• Disallowed
• Leads to increased search
• Limits the power of the constraint programming to
  use dynamic variable ordering heuristics

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Lex-Leader Limitation

• Requires one constraint for each element of the
  symmetry group
• Many groups contain an exponential number of
  symmetries
• In the case of matrix with m rows and n columns,
  this is m!n!
• Applicable to many cases but it is impractical