5.277. roots

DESCRIPTIONLINKSGRAPH
Origin

[BessiereHebrardHnichKiziltanWalsh05IJCAI]

Constraint

πš›πš˜πš˜πšπšœ(πš‚,πšƒ,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚)

Arguments
πš‚πšœπšŸπšŠπš›
πšƒπšœπšŸπšŠπš›
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›-πšπšŸπšŠπš›)
Restrictions
πš‚β‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšŸπšŠπš›)
Purpose

πš‚ is the set of indices of the variables in the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ taking their values in πšƒ; πš‚={i | πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›βˆˆπšƒ}. Positions are numbered from 1.

Example
{2,4,5},{2,3,8},1,3,1,2,3

The πš›πš˜πš˜πšπšœ constraint holds since values 2 and 3 in πšƒ occur in the collection 〈1,3,1,2,3βŒͺ only at positions πš‚={2,4,5}. The value 8βˆˆπšƒ does not occur within the collection 〈1,3,1,2,3βŒͺ.

Usage

BessiΓ¨re et al. showedΒ [BessiereHebrardHnichKiziltanWalsh05IJCAI] that many counting and occurence constraints can be specified with two global primitives: πš›πš˜πš˜πšπšœ and πš›πšŠπš—πšπšŽ. For instance, the πšŒπš˜πšžπš—πš constraint can be decomposed into one πš›πš˜πš˜πšπšœ constraint: πšŒπš˜πšžπš—πš(πš…π™°π™»,πš…π™°πšπš‚,𝙾𝙿,π™½πš…π™°πš) iff πš›πš˜πš˜πšπšœ(πš‚,{πš…π™°π™»},πš…π™°πšπš‚)∧|πš‚| 𝙾𝙿 π™½πš…π™°πš.

πš›πš˜πš˜πšπšœ does not count but collects the set of variables using particular values. It provides then a way of channeling. πš›πš˜πš˜πšπšœ generalizes, for instance, the πš•πš’πš—πš”_𝚜𝚎𝚝_𝚝𝚘_πš‹πš˜πš˜πš•πšŽπšŠπš—πšœ constraint, πš•πš’πš—πš”_𝚜𝚎𝚝_𝚝𝚘_πš‹πš˜πš˜πš•πšŽπšŠπš—πšœ(πš‚,π™±π™Ύπ™Ύπ™»π™΄π™°π™½πš‚) iff πš›πš˜πš˜πšπšœ(πš‚,{1},π™±π™Ύπ™Ύπ™»π™΄π™°π™½πš‚.πš‹πš˜πš˜πš•), or may be used instead of the πšπš˜πš–πšŠπš’πš—_πšŒπš˜πš—πšœπšπš›πšŠπš’πš—πš.

Other examples of reformulations are given inΒ [BessiereHebrardHnichKiziltanWalsh09].

Algorithm

InΒ [BessiereHebrardHnichKiziltanWalsh06CP], BessiΓ¨re et al. shows that enforcing hybrid-consistency on πš›πš˜πš˜πšπšœ is NP -hard. They consider the decomposition of πš›πš˜πš˜πšπšœ into a network of ternary constraints: βˆ€i, iβˆˆπš‚β‡’πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›βˆˆπšƒ and πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›β‡’πšƒβˆ§iβˆˆπš‚. Enforcing bound consistency on the decomposition achieves bound consistency on πš›πš˜πš˜πšπšœ. Enforcing hybrid consistency on the decomposition achieves at least bound consistency on πš›πš˜πš˜πšπšœ, until hybrid consistency in some special cases:

  • π‘‘π‘œπ‘š(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›)βŠ‚πšƒ Μ², βˆ€iβˆˆπš‚ Μ²,

  • π‘‘π‘œπ‘š(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›)βˆ©πšƒ Β―=βˆ…, βˆ€iβˆ‰πš‚ Β―,

  • πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are ground,

  • πšƒ is ground.

Enforcing hybrid consistency on the decomposition can be done in O(nd) with n=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚| and d the maximum domain size of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš› and πšƒ.

See also

common keyword: πš•πš’πš—πš”_𝚜𝚎𝚝_𝚝𝚘_πš‹πš˜πš˜πš•πšŽπšŠπš—πšœΒ (constraint involving set variables).

related: πšŠπš–πš˜πš—πšΒ (can be expressed with πš›πš˜πš˜πšπšœ), πšŠπšœπšœπš’πšπš—_πšŠπš—πš_πš—πšŸπšŠπš•πšžπšŽπšœΒ (can be expressed with πš›πš˜πš˜πšπšœ and πš›πšŠπš—πšπšŽ), πšŠπšπš•πšŽπšŠπšœπš, πšŠπšπš–πš˜πšœπšΒ (can be expressed with πš›πš˜πš˜πšπšœ), πšŒπš˜πš–πš–πš˜πš—Β (can be expressed with πš›πš˜πš˜πšπšœ and πš›πšŠπš—πšπšŽ), πšŒπš˜πšžπš—πšΒ (can be expressed with πš›πš˜πš˜πšπšœ), πšπš˜πš–πšŠπš’πš—_πšŒπš˜πš—πšœπšπš›πšŠπš’πš—πš, πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’, πšπš•πš˜πš‹πšŠπš•_πšŒπš˜πš—πšπš’πšπšžπš’πšπš’Β (can be expressed with πš›πš˜πš˜πšπšœ), πšœπš’πš–πš–πšŽπšπš›πš’πšŒ_πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš, 𝚞𝚜𝚎𝚜 (can be expressed with πš›πš˜πš˜πšπšœ and πš›πšŠπš—πšπšŽ).

Keywords

characteristic of a constraint: disequality.

constraint arguments: constraint involving set variables.

constraint type: counting constraint, value constraint, decomposition.

filtering: hybrid-consistency.

Derived Collection
πšŒπš˜πš•(πš‚π™΄πšƒπš‚-πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚜-πšœπšŸπšŠπš›,𝚝-πšœπšŸπšŠπš›),[πš’πšπšŽπš–(𝚜-πš‚,𝚝-πšƒ)])
Arc input(s)

πš‚π™΄πšƒπš‚ πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

Arc generator
π‘ƒπ‘…π‘‚π·π‘ˆπΆπ‘‡β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚜𝚎𝚝𝚜,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ)

Arc arity
Arc constraint(s)
πš’πš—_𝚜𝚎𝚝(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ.πš”πšŽπš’,𝚜𝚎𝚝𝚜.𝚜)β‡”πš’πš—_𝚜𝚎𝚝(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ.πšŸπšŠπš›,𝚜𝚎𝚝𝚜.𝚝)
Graph property(ies)
𝐍𝐀𝐑𝐂=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|

Graph model
Figure 5.277.1. Initial and final graph of the πš›πš˜πš˜πšπšœ constraint
ctrs/rootsActrs/rootsB
(a) (b)