5.348. used_by_partition

DESCRIPTIONLINKSGRAPH
Origin

Derived from 𝚞𝚜𝚎𝚍_πš‹πš’.

Constraint

𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš—(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)

Type
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šπ™Έπ™°π™±π™»π™΄πš‚1π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šπ™Έπ™°π™±π™»π™΄πš‚2π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™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™°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™°πšπ™Έπ™°π™±π™»π™΄πš‚1|β‰₯|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1,πšŸπšŠπ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‚2,πšŸπšŠπš›)
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

For each integer i in [1,|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|], let 𝑁1 i (respectively 𝑁2 i ) denote the number of variables of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 (respectively πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2) that take their value in the i th partition of the collection π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚. For all i in [1,|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|] we have 𝑁2 i >0⇒𝑁1 i >0.

Example
πšŸπšŠπš›-1,πšŸπšŠπš›-9,πšŸπšŠπš›-1,πšŸπšŠπš›-6,πšŸπšŠπš›-2,πšŸπšŠπš›-3,1,3,6,6,πš™-1,3,πš™-4,πš™-2,6

The different values of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2=〈1,3,6,6βŒͺ are respectively associated with the partitions πš™-〈1,3βŒͺ, πš™-〈1,3βŒͺ, πš™-〈2,6βŒͺ, and πš™-〈2,6βŒͺ. Therefore partitions πš™-〈1,3βŒͺ and πš™-〈2,6βŒͺ are respectively used 2 and 2 times.

Similarly, the different values of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1=〈1,9,1,6,2,3βŒͺ (except value 9, which does not occur in any partition) are respectively associated with the partitions πš™-〈1,3βŒͺ, πš™-〈1,3βŒͺ, πš™-〈2,6βŒͺ, πš™-〈2,6βŒͺ, and πš™-〈1,3βŒͺ. Therefore partitions πš™-〈1,3βŒͺ and πš™-〈2,6βŒͺ are respectively used 3 and 2 times.

Consequently, the 𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint holds since, for each partition associated with the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2=〈1,3,6,6βŒͺ, its number of occurrences within πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1=〈1,9,1,6,2,3βŒͺ is greater than or equal to its number of occurrences within πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2:

  • Partition πš™-〈1,3βŒͺ occurs 3 times within 〈1,9,1,6,2,3βŒͺ and 2 times within 〈1,3,6,6βŒͺ.

  • Partition πš™-〈2,6βŒͺ occurs 2 times within 〈1,9,1,6,2,3βŒͺ and 2 times within 〈1,3,6,6βŒͺ.

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

  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 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šπ™Έπ™°π™±π™»π™΄πš‚1.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‚.

  • An occurrence of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.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‚.

Used in

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

See also

implied by: πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

soft variant: 𝚜𝚘𝚏𝚝_𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš—_πšŸπšŠπš›Β (variable-based violation measure).

specialisation: 𝚞𝚜𝚎𝚍_πš‹πš’Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽβˆˆπš™πšŠπš›πšπš’πšπš’πš˜πš— replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

system of constraints: πš”_𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

used in graph description: πš’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—.

Keywords

characteristic of a constraint: partition.

constraint arguments: constraint between two collections of variables.

modelling: inclusion.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2

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—(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š’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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›,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)
Graph property(ies)
β€’ for all connected components: ππ’πŽπ”π‘π‚π„β‰₯ππ’πˆππŠ
β€’ ππ’πˆππŠ=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.348.1 respectively show the initial and final graph associated with the Example slot. Since we use the ππ’πŽπ”π‘π‚π„ and ππ’πˆππŠ graph properties, the source and sink vertices of the final graph are stressed with a double circle. Since there is a constraint on each connected component of the final graph we also show the different connected components. Each of them corresponds to an equivalence class according to the arc constraint. Note that the vertex corresponding to the variable that takes value 9 was removed from the final graph since there is no arc for which the associated equivalence constraint holds. The 𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint holds since:

  • For each connected component of the final graph the number of sources is greater than or equal to the number of sinks.

  • The number of sinks of the final graph is equal to |πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|.

Figure 5.348.1. Initial and final graph of the 𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint
ctrs/used_by_partitionA
(a)
ctrs/used_by_partitionB
(b)
Signature

Since the initial graph contains only sources and sinks, and since sources of the initial graph cannot become sinks of the final graph, we have that the maximum number of sinks of the final graph is equal to |πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|. Therefore we can rewrite ππ’πˆππŠ=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2| to ππ’πˆππŠβ‰₯|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2| and simplify ππ’πˆππŠ Β― Μ² to ππ’πˆππŠ Β―.