5.181. k_same_partition

DESCRIPTIONLINKSGRAPH
Origin

Derived from πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— and from πš”_πšœπšŠπš–πšŽ.

Constraint

πš”_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—(πš‚π™΄πšƒπš‚,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)

Types
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›)
πš…π™°π™»πš„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š—πš)
Arguments
πš‚π™΄πšƒπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚜𝚎𝚝-πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚)
π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš™-πš…π™°π™»πš„π™΄πš‚)
Restrictions
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›)
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>0
|πš…π™°π™»πš„π™΄πš‚|β‰₯1
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
πšπš’πšœπšπš’πš—πšŒπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
πš›πšŽπššπšžπš’πš›πšŽπš(πš‚π™΄πšƒπš‚,𝚜𝚎𝚝)
|πš‚π™΄πšƒπš‚|>1
πšœπšŠπš–πšŽ_πšœπš’πš£πšŽ(πš‚π™΄πšƒπš‚,𝚜𝚎𝚝)
πš›πšŽπššπšžπš’πš›πšŽπš(π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚,πš™)
|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|β‰₯2
Purpose

Given a collection of |πš‚π™΄πšƒπš‚| sets, each containing the same number of domain variables, the πš”_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint enforces a πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint between each pair of consecutive sets.

Example
𝚜𝚎𝚝-πšŸπšŠπš›-1,πšŸπšŠπš›-2,πšŸπšŠπš›-6,πšŸπšŠπš›-3,πšŸπšŠπš›-1,πšŸπšŠπš›-2,𝚜𝚎𝚝-πšŸπšŠπš›-6,πšŸπšŠπš›-6,πšŸπšŠπš›-2,πšŸπšŠπš›-3,πšŸπšŠπš›-1,πšŸπšŠπš›-3,𝚜𝚎𝚝-πšŸπšŠπš›-2,πšŸπšŠπš›-2,πšŸπšŠπš›-2,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πš™-1,3,πš™-4,πš™-2,6

The first argument πš‚π™΄πšƒπš‚ of the πš”_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint corresponds to 3 collections of variables, while the second argument π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ defines the 3 sets of values {1,3}, {4} and {2,6}. The πš”_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint holds since:

  • The first and second collections of variables are assigned 3 values in the {1,3} as well as 3 values in {2,6}.

  • The second and third collections of variables are also assigned 3 values in the {1,3} as well as 3 values in {2,6}.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>1
Symmetries
  • Items of πš‚π™΄πšƒπš‚ are permutable.

  • Items of πš‚π™΄πšƒπš‚.𝚜𝚎𝚝 are permutable.

  • Items of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ are permutable.

  • Items of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.πš™ are permutable.

  • An occurrence of a value of πš‚π™΄πšƒπš‚.𝚜𝚎𝚝.πšŸπšŠπš› can be replaced by any other value that also belongs to the same partition of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

See also

common keyword: πš”_πšœπšŠπš–πšŽΒ (system of constraints).

implies: πš”_𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

part of system of constraints: πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

used in graph description: πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

Keywords

characteristic of a constraint: partition.

combinatorial object: permutation.

constraint type: system of constraints, decomposition.

Arc input(s)

πš‚π™΄πšƒπš‚

Arc generator
π‘ƒπ΄π‘‡π»β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚜𝚎𝚝1,𝚜𝚎𝚝2)

Arc arity
Arc constraint(s)
πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—(𝚜𝚎𝚝1.𝚜𝚎𝚝,𝚜𝚎𝚝2.𝚜𝚎𝚝,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)
Graph property(ies)
𝐍𝐀𝐑𝐂=|πš‚π™΄πšƒπš‚|-1

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.181.1 respectively show the initial and final graph associated with the Example slot. To each vertex corresponds a collection of variables, while to each arc corresponds a πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint.

Figure 5.181.1. Initial and final graph of the πš”_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint
ctrs/k_same_partitionActrs/k_same_partitionB
(a) (b)