Consistency

1
Consistency Propagation with Multiset
Constraints

• Toby Walsh
• 4C, UCC Uppsala

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Outline

• Multiset vars
• vars which take a multiset (or bag) of values
• Representing multiset vars
• Consistency of multiset vars
• Global constraints over multiset vars
• Conclusions

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Multiset variables

• Assigned a multiset or bag of values
• Useful in design configuration
• E.g. template design (prob002 in CSPLib)
• Multiset of labels on each printing template
• tuna, tuna, tuna, chicken, liver, liver

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Multiset variables

• Assigned a multiset or bag of values
• Useful in design configuration
• E.g. ILOGs Configurator
• Multiset of components in home cinema system
• plasma screen, DVD, amplifier, speaker,
  speaker, speaker, speaker, woofer

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Representing multiset vars

• Naïve, explicit representation
• Finite domain of all possible multisets
• BUT exponential
• More compact (but less expressive)
• Bounds
• Occurrences
• Fixed cardinality

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Bounds representation

• Upper and lower bounds on multiset
• Lower bound must contain
• Upper bound can contain
• E.g lb is DVD, speaker, speaker
• ub is DVD, plasma, TV, satellite dish,
  amplifier, speaker, speaker, speaker, speaker,
  woofer

Similar bounds representation used for set vars
in Gervet 97
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Occurrence representation

• Integer variable representing occurrences of
  each value
• E.g. DVD 1, Plasma0,1, Speaker2,3,4,
  Woofer0,1,

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Fixed cardinality representation

• For multisets with fixed (or bounded) cardinality
• Finite-domain variable for each element in the
  multiset
• Introduce dont care value if needed
• E.g. M1M2M3 Video, DVD, plasma, TV,
  speaker, woofer, satellite

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Expressivity

• Disjunctive choice cannot be fully expressed
• E.g. none of the 3 can represent just
• 0,0,0 or 1,1,1
• Nested multisets are also problematic
• Especially for occurrence representation

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Consistency

• Central notion in CP
• How do we define it for multiset vars?
• Also for mixed constraints containing multiset,
  set and/or integer vars?

Slogan of the local Beamish brewery Consistency
in a world gone mad
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Consistency

• Set of solutions for a var in a constraint C
• Sol(Xi) di C(d1,..,dn)
• C is bounds consistent iff
• Sol(Xi) ?
• For each (multi)set var
• lub(Xi)? di, glb(Xi)? di where di ?
  Sol(Xi)
• For each integer var
• lub(Xi)max(di), glb(Xi)min(di) where di
  ? Sol(Xi)

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Bounds consistency

• Representation doesnt matter
• BC on bounds representation is equivalent to
  usual notion of BC on occurrence
• Existence of BC domains
• If problem has solution, there exist unique lub
  and glb that makes constraints BC

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Multiset constraints

• Simple predicates
• X?Y, X?Y, XY, X?Y, XN, occ(N,X)M,
• Arguments can be multiset or set expressions
• Var, ground multiset or set
• X?Y, X?Y, X-Y,
• Global constraints
• disjoint(X1,..,Xn), partition(X1,,Xn,X),

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Normal form

• Multiset constraints can be put in simple normal
  form
• Each constraint is at most ternary
• E.g. (X?Y)?Z ? W is transformed to
• SX?Y, TS?Z, T ? W
• Propagation algorithms therefore only need deal
  with a limited class of constraints

Similar to normal form used with sets vars in
Gervet 97
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Normal form

• Normalization hurts propagation
• BC on arbitrary set of constraints is strictly
  stronger than BC on normalized form
• But not always
• BC on normalized form is equivalent if there are
  no repeated variables

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Global constraints

• Important aspect of (finite-domain) CP
• Identify common patterns
• Efficient and effective propagators
• Decomposition usually hurts
• E.g. all-different clique of not-equals
• GAC(all-different) gt AC(not-equals)

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Global multiset constraints

• Decomposition does not always hurt
• BC(disjoint(X1,..,Xn) BC(Xi ? Xj)
• BC(partition(X1,..,Xn,X)BC(Xi ? Xj)
  BC(Xi ? .. ? Xn X)
• But can on other global constraints
• non-empty partition
• distinct(X1,..,Xn) iff Xi ? Xj for all iltj

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Future directions

• Multiset OPL
• Extend OPL to support set and multiset vars
• Compile down into OPL using occurrence or fixed
  cardinality representations (or both)
• Global constraint propagators
• distinct, non-empty partition,

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What do you take home?

• Representing multiset vars
• Bounds, occurrence and fixed cardinality
• Consistency of multiset vars
• Containing multiset, set and integer vars
• Global constraints over multiset vars
• Decomposition does not always hurt

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Whats the bigger picture?

• Why stop with multisets?
• Strings Golden Pang, CP03 Fri 10.15
• Sequences
• Trees
• Why should CP have only fixed variable types?
• Abstract variable types