Consistency & Propagation with Multiset Constraints

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 • 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

Consistency & Propagation with Multiset Constraints