Specialised Binary Constraint for the Stable Marriage Problem

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Specialised Binary Constraint for the Stable
Marriage Problem

• By Chris Unsworth and Patrick Prosser

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Contents

• The Stable Marriage Problem
• The Algorithm
• Previous Constraint Models
• Specialised Binary Constraint
• Computational Comparison
• Conclusion
• Applause

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Contents

• The Stable Marriage Problem
• The Algorithm
• Previous Constraint Models
• Specialised Binary Constraint
• Computational Comparison
• Conclusion
• Applause

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The Stable Marriage Problem
We have n men
and n women
Each man ranks the n women
And each woman ranks the n men
Men
Women
Ian Jon Bob Jon Ian Bob Bob Jon Ian
Jan Liz Sue
Bob Ian Jon
Sue Jan Liz Liz Jan Sue Jan Sue Liz
Objective To find a matching of men to women
Such that the matching is Stable
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The Stable Marriage Problem
A Matching
But not a stable one ?
Bob and Sue would rather be matched to each other
than to their assigned partners
Men
Women
Ian Jon Bob Jon Ian Bob Bob Jon Ian
Jan Liz Sue
Bob Ian Jon
Sue Jan Liz Liz Jan Sue Jan Sue Liz
In this matching Bob and Sue are a Blocking pair
A matching is only stable iff it contains no
Blocking pairs
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The Stable Marriage Problem
A Stable Matching
And another Stable Match
Men
Women
Ian Jon Bob Jon Ian Bob Bob Jon Ian
Jan Liz Sue
Bob Ian Jon
Sue Jan Liz Liz Jan Sue Jan Sue Liz
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Contents

• The Stable Marriage Problem
• The Algorithm
• Previous Constraint Models
• Specialised Binary Constraint
• Computational Comparison
• Conclusion
• Applause

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The Extended Gale Shapley Algorithm

• Cycles through a list of free men until there are
  no free men remaining
• The man proposes to his most preferred woman
• If the woman was engaged then the engagement is
  broken and her previous fiancé will be added to
  the free list
• The man and woman become engaged
• All men the woman likes less than her new fiancé
  will be removed from her preference list, and she
  will also be removed from theirs

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The Extended Gale Shapley Algorithm
assign each person to be free WHILE (some man m
is free) DO BEGIN w first woman on m's
list IF (some man p is engaged to w) THEN
assign p to be free assign m and w to be engaged
to each other FOR (each successor p of m on w's
list) DO BEGIN delete p from w's list delete
w from p's list END END
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The Extended Gale Shapley Algorithm

• The reduced preference lists are the MGS-Lists
• The female version yields the WGS-Lists
• The intersection of these lists are known as the
  GS-Lists
• Contains all possible stable matchings
• If all men are match to there best match in the
  GS-Lists that matching will be the man optimal
  matching
• Likewise with the women

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Contents

• The Stable Marriage Problem
• The Algorithm
• Previous Constraint Models
• Specialised Binary Constraint
• Computational Comparison
• Conclusion
• Applause

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Previous Constraint Models

• Two Constraint encodings presented by Ian Gent,
  Robert Irving, David Manlove, Patrick Prosser and
  Barbara Smith in CP01

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Forbidden tuples model (CP01)

• 2n variables, one for each man and woman each
  with a domain (1 .. N)
• n2 table constraints
• One for each potential couple
• Set of pairs of values the constrained variables
  cannot simultaneously take
• Potentially O(n2) pairs per constraint

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Forbidden tuples model (CP01)

• When made arc consistent the variable domains are
  equivalent to the GS-Lists
• All stable matchings can be found in a failure
  free enumeration
• Space complexity O(n4)
• Time complexity O(n4)

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Boolean Encoding (CP01)

• 2n2 variables with 0/1 domains
• mi,j 1 means mi is married to a partner no
  better than j.
• mi,j 0 means mi is married to a partner
  better than j
• O(n2) constraints, mostly implication

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Boolean Encoding (CP01)

• When made arc consistent the variable domains
  equal the bounds of the GS-lists
• All stable matchings can be found in a failure
  free enumeration
• Space complexity O(n2)
• Time complexity O(n2)

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Contents

• The Stable Marriage Problem
• The Algorithm
• Previous Constraint Models
• Specialised Binary Constraint
• Computational Comparison
• Conclusion
• Applause

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Justification

• A Linear time algorithm already exists for this
  problem (linear to the size of input i.e. O(n2))
• Optimal Constraint models have been published
• So why do we need a specialised constraint for
  this problem?
• Performance gap between Constraint models and
  algorithm
• Inflexibility of the algorithm
• The Aim of the Specialised Constraint Is then to
  reduce the performance gap whilst retaining the
  flexibility of the constraint models

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Specialised Binary Constraint

• 2n variables, one for each man and woman each
  with a domain (1 .. N)
• Domain values represent preferences
• e.g. if the man variable m1 were assigned the
  value 3 then man 1 would be assigned to his third
  choice woman
• n2 Specialised Binary Constraints
• One for each potential couple

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Specialised Binary Constraint

• 5 methods associated with a constraint
• Initialisation of constraints
• Lower bound of a variable increases
• Upper bound of a variable increases
• Interior value removal
• Variable instantiation
• Assume constraint is processed in an AC5-like
  environment
• when a variable loses a value, process the
  constraints (n of these) incident on that
  variable

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Specialised Binary Constraint
mi
wj

• Constraint between mi and wj
• Access to mis domain
• Access to wjs domain
• Position of wj in mis preference list
• Position of mi in wjs preference list

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Lower bound increases
mi
wj

• Man mis lower bound increases
• The Lower bound method is called
• Man mis new favourite is wj
• Woman wj will then reject all men she likes less
  than mi
• Woman wjs upper bound is reduced

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Upper bound decreases
wj
mi

• Woman wjs upper bound is reduced
• The upper bound method is called
• Man mi is no longer in wjs domain
• Woman wj is removed from mis domain

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Initialisation, Interior Removal, Instantiation

• Initialisation simply calls the lower bound
  method
• The Interior Removal and Instantiation methods
  are required for search and side constraints
• Read paper for more detail

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An Inefficiency

• Revision of O(n) constraints when only one of
  these will have an effect
• Space complexity O(n2)
• Time complexity O(n3)
• Due to redundant calls above
• See paper for complexity argument

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Contents

• The Stable Marriage Problem
• The Algorithm
• Previous Constraint Models
• Specialised Binary Constraint
• Computational Comparison
• Conclusion
• Applause

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Computational Comparison

• Compare forbidden tuples (FT), Boolean (Bool),
  and the specialised binary constraint (SM2) on
  random SMP
• Time to make arc-consistent
• Time to find all solutions
• All times include model creation time

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Computational Comparison
Time in seconds to produce the GS-Lists
Time in seconds to find all solutions
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Computational Comparison

• Why is suboptimal model dominating?
• SM2 is more space efficient
• Bool has 2n 6n2 constraints
• SM2 has n2 constraints
• Bool has 2n2 variables
• SM2 has 2n variables

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Demonstration of Versatility

• The sex equal stable marriage problem
• Find the matching that is equally good for all
• Minimise the absolute difference between the sums
  of the preferences of the male and female
  assignment
• Minimise( )
• Where P(m1) gives m1s preference for his
  assigned partner
• This has been proven to be NP-hard

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Demonstration of Versatility

• Implementation
• 2 summation constraints
• 1 over the male variable the other over the
  female
• 1 variable to hold the value of the absolute
  difference between the 2 summation constraints
• 1 minimising objective constraint
• Code needed

final IlcIntVar optVar solver.intVar(0,
nn) solver.add(solver.eq(optVar, solver.abs(
solver.diff(solver.sum(menVars),
solver.sum(womenVars))))) solver.add(solver.mini
mize(optVar))
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Demonstration of Versatility
Time in seconds to find sex-equal optimal
solution
Time in seconds to find all solutions
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Conclusion

• A new specialised binary constraint for SM
• Outperforms optimal constraint model
• Because compact and simple
• Versatile
• 1st computational study of SESMP (we think)
• Room for improvement
• Avoid redundant revisions (see The Fifth Workshop
  on Modelling and Solving Problems with
  Constraints at IJCAI05)

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Applause Questions

• Your turn ?