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Consistency & Propagation with Multiset Constraints
Toby Walsh4C, UCC & Uppsala
Outline
Multiset varsvars which take a multiset (or bag) of values
Representing multiset vars Consistency of multiset vars Global constraints over multiset vars Conclusions
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}}
Multiset variables
Assigned a multiset or bag of values Useful in design & configuration
E.g. ILOG’s ConfiguratorMultiset of components in home cinema system: {{plasma screen, DVD, amplifier, speaker, speaker, speaker, speaker, woofer}}
Representing multiset vars
Naïve, explicit representation Finite domain of all possible multisets BUT exponential
More compact (but less expressive) Bounds Occurrences Fixed cardinality
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 representationused for set vars in [Gervet 97]
Occurrence representation
Integer variable representing #occurrences of each value
E.g. DVD = {1}, Plasma={0,1}, Speaker={2,3,4}, Woofer={0,1}, …
Fixed cardinality representation
For multisets with fixed (or bounded) cardinality
Finite-domain variable for each element in the multiset Introduce “don’t care” value if needed E.g. M1=M2=M3= … = {Video, DVD, plasma,
TV, speaker, woofer, satellite}
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
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”
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)
Bounds consistency
Representation doesn’t 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
Multiset constraints
Simple predicates XY, XY, X=Y, XY, |X|=N, occ(N,X)=M, … Arguments can be multiset or set expressions
Var, ground multiset or set XY, XY, X-Y, …
Global constraints disjoint(X1,..,Xn), partition(X1,…,Xn,X), …
Normal form
Multiset constraints can be put in simple normal form Each constraint is at most ternary E.g. (XY)Z W is transformed to
S=XY, T=SZ, 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]
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
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) > AC(not-equals)
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 i<j
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, …
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
What’s 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