5.240. nvalues

DESCRIPTIONLINKSGRAPH
Origin

Inspired by πš—πšŸπšŠπš•πšžπšŽ and πšŒπš˜πšžπš—πš.

Constraint

πš—πšŸπšŠπš•πšžπšŽπšœ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšπ™΄π™»π™Ύπ™Ώ,π™»π™Έπ™Όπ™Έπšƒ)

Arguments
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™Ύπ™ΏπšŠπšπš˜πš–
π™»π™Έπ™Όπ™ΈπšƒπšπšŸπšŠπ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™»π™Ύπ™Ώβˆˆ[=,β‰ ,<,β‰₯,>,≀]
Purpose

Let N be the number of distinct values assigned to the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection. Enforce condition N πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ to hold.

Example
πšŸπšŠπš›-4,πšŸπšŠπš›-5,πšŸπšŠπš›-5,πšŸπšŠπš›-4,πšŸπšŠπš›-1,πšŸπšŠπš›-5,=,3

The πš—πšŸπšŠπš•πšžπšŽπšœ constraint holds since the number of distinct values occurring within the collection 〈4,5,5,4,1,5βŒͺ is equal (i.e.,Β πšπ™΄π™»π™Ύπ™Ώ is set to =) to its third argument π™»π™Έπ™Όπ™Έπšƒ=3.

Symmetries
  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are permutable.

  • All occurrences of two distinct values of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› can be swapped; all occurrences of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› can be renamed to any unused value.

Usage

Used in the Constraint(s) on sets slot for defining some constraints like πšŠπšœπšœπš’πšπš—_πšŠπš—πš_πš—πšŸπšŠπš•πšžπšŽπšœ, πšŒπš’πš›πšŒπšžπš’πš_πšŒπš•πšžπšœπšπšŽπš› or πšŒπš˜πš•πš˜πšžπš›πšŽπš_πšŒπšžπš–πšžπš•πšŠπšπš’πšŸπšŽ.

Reformulation

The πš—πšŸπšŠπš•πšžπšŽπšœ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚, πšπ™΄π™»π™Ύπ™Ώ ,π™»π™Έπ™Όπ™Έπšƒ) constraint can be expressed in term of the conjunction πš—πšŸπšŠπš•πšžπšŽ(𝑁𝑉,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚) ∧ 𝑁𝑉 πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ.

Used in

πšŠπšœπšœπš’πšπš—_πšŠπš—πš_πš—πšŸπšŠπš•πšžπšŽπšœ, πšŒπš’πš›πšŒπšžπš’πš_πšŒπš•πšžπšœπšπšŽπš›, πšŒπš˜πš•πš˜πšžπš›πšŽπš_πšŒπšžπš–πšžπš•πšŠπšπš’πšŸπšŽ, πšŒπš˜πš•πš˜πšžπš›πšŽπš_πšŒπšžπš–πšžπš•πšŠπšπš’πšŸπšŽπšœ.

See also

assignment dimension added: πšŠπšœπšœπš’πšπš—_πšŠπš—πš_πš—πšŸπšŠπš•πšžπšŽπšœ.

common keyword: πš—πšŸπšŠπš•πšžπšŽπšœ_πšŽπš‘πšŒπšŽπš™πš_0Β (counting constraint,number of distinct values).

specialisation: πš—πšŸπšŠπš•πšžπšŽΒ (replace a comparison with the number of distinct values by an equality with the number of distinct values).

Keywords

constraint type: counting constraint, value partitioning constraint.

final graph structure: strongly connected component, equivalence.

modelling: number of distinct equivalence classes, number of distinct values.

problems: domination.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

Arc generator
πΆπΏπΌπ‘„π‘ˆπΈβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2)

Arc arity
Arc constraint(s)
πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš›=πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›
Graph property(ies)
𝐍𝐒𝐂𝐂 πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ

Graph class
π™΄πš€πš„π™Έπš…π™°π™»π™΄π™½π™²π™΄

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.240.1 respectively show the initial and final graph associated with the Example slot. Since we use the 𝐍𝐒𝐂𝐂 graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a value that is assigned to some variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection. The 3 following values 1, 4 and 5 are used by the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection.

Figure 5.240.1. Initial and final graph of the πš—πšŸπšŠπš•πšžπšŽπšœ constraint
ctrs/nvaluesActrs/nvaluesB
(a) (b)