5.243. nvectors

DESCRIPTIONLINKSGRAPH
Origin

Inspired by πš—πšŸπšŽπšŒπšπš˜πš› and πšŒπš˜πšžπš—πš.

Constraint

πš—πšŸπšŽπšŒπšπš˜πš›πšœ(πš…π™΄π™²πšƒπ™Ύπšπš‚,πšπ™΄π™»π™Ύπ™Ώ,π™»π™Έπ™Όπ™Έπšƒ)

Type
πš…π™΄π™²πšƒπ™Ύπ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›)
Arguments
πš…π™΄π™²πšƒπ™Ύπšπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚟𝚎𝚌-πš…π™΄π™²πšƒπ™Ύπ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™²πšƒπ™Ύπšπš‚,𝚟𝚎𝚌)
πšœπšŠπš–πšŽ_πšœπš’πš£πšŽ(πš…π™΄π™²πšƒπ™Ύπšπš‚,𝚟𝚎𝚌)
πšπ™΄π™»π™Ύπ™Ώβˆˆ[=,β‰ ,<,β‰₯,>,≀]
Purpose

Let N be the number of distinct tuples of values taken by the vectors of the πš…π™΄π™²πšƒπ™Ύπšπš‚ collection. Enforce condition N πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ to hold.

Example
𝚟𝚎𝚌-5,6,𝚟𝚎𝚌-5,6,𝚟𝚎𝚌-9,3,𝚟𝚎𝚌-5,6,𝚟𝚎𝚌-9,3,=,2

The πš—πšŸπšŽπšŒπšπš˜πš›πšœ constraint holds since the number of distinct tuples of values (i.e.,Β tuples 〈5,6βŒͺ and 〈9,3βŒͺ) occurring within the collection πš…π™΄π™²πšƒπ™Ύπšπš‚ 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=2.

Symmetries
  • Items of πš…π™΄π™²πšƒπ™Ύπšπš‚ are permutable.

  • Items of πš…π™΄π™²πšƒπ™Ύπšπš‚.𝚟𝚎𝚌 are permutable (same permutation used).

  • All occurrences of two distinct values of πš…π™΄π™²πšƒπ™Ύπšπš‚.𝚟𝚎𝚌 can be swapped; all occurrences of a value of πš…π™΄π™²πšƒπ™Ύπšπš‚.𝚟𝚎𝚌 can be renamed to any unused value.

Reformulation

The πš—πšŸπšŽπšŒπšπš˜πš›πšœ(πš…π™΄π™²πšƒπ™Ύπš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šŽπšŒπšπš˜πš›(𝑁𝑉,πš…π™΄π™²πšƒπ™Ύπšπš‚) ∧ 𝑁𝑉 πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ.

See also

specialisation: πš—πšŸπšŽπšŒπšπš˜πš›Β (replace a comparison with the number of distinct vectors by an equality with the number of distinct vectors).

Keywords

characteristic of a constraint: vector.

constraint type: counting constraint, value partitioning constraint.

final graph structure: strongly connected component, equivalence.

modelling: number of distinct equivalence classes.

problems: domination.

Arc input(s)

πš…π™΄π™²πšƒπ™Ύπš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šŽπšŒπšπš˜πš›πšœ1,πšŸπšŽπšŒπšπš˜πš›πšœ2)

Arc arity
Arc constraint(s)
πš•πšŽπš‘_πšŽπššπšžπšŠπš•(πšŸπšŽπšŒπšπš˜πš›πšœ1.𝚟𝚎𝚌,πšŸπšŽπšŒπšπš˜πš›πšœ2.𝚟𝚎𝚌)
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 5.243.1 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 tuple of values that is assigned to some vectors of the πš…π™΄π™²πšƒπ™Ύπšπš‚ collection. The 2 following tuple of values 〈5,6βŒͺ and 〈9,3βŒͺ are used by the vectors of the πš…π™΄π™²πšƒπ™Ύπšπš‚ collection.

Figure 5.243.1. Initial and final graph of the πš—πšŸπšŽπšŒπšπš˜πš›πšœ constraint
ctrs/nvectorsActrs/nvectorsB
(a) (b)