Consistency & Propagation with
Multiset Constraints
Toby Walsh
4C, UCC & Uppsala
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
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 Configurator
Multiset 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 representation
used 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


XY, XY, X=Y, XY, |X|=N, 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), …
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
S=XY, T=S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]
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


Consistency of multiset vars


Bounds, occurrence and
fixed cardinality
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