5.279. same_and_global_cardinality

DESCRIPTIONLINKSGRAPH
Origin

Conjoin πšœπšŠπš–πšŽ and 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’

Constraint

πšœπšŠπš–πšŽ_πšŠπ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™°πšπ™Έπ™°π™±π™»π™΄πš‚1,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2,πš…π™°π™»πš„π™΄πš‚)

Synonyms

𝚜𝚐𝚌𝚌, πšœπšŠπš–πšŽ_𝚐𝚌𝚌, πšœπšŠπš–πšŽ_πšŠπš—πš_𝚐𝚌𝚌, 𝚜𝚠𝚌, πšœπšŠπš–πšŽ_πš πš’πšπš‘_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’πšŽπšœ.

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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,πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-πšπšŸπšŠπš›)
Restrictions
|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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š…π™°π™»πš„π™΄πš‚,[πšŸπšŠπš•,πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ])
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™»πš„π™΄πš‚.πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽβ‰₯0
πš…π™°π™»πš„π™΄πš‚.πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽβ‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|
Purpose

The variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 collection correspond to the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 collection according to a permutation. In addition, each value πš…π™°π™»πš„π™΄πš‚[i].πšŸπšŠπš• (1≀i≀|πš…π™°π™»πš„π™΄πš‚|) should be taken by exactly πš…π™°π™»πš„π™΄πš‚[i].πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 collection.

Example
πšŸπšŠπš›-1,πšŸπšŠπš›-9,πšŸπšŠπš›-1,πšŸπšŠπš›-5,πšŸπšŠπš›-2,πšŸπšŠπš›-1,πšŸπšŠπš›-9,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-2,πšŸπšŠπš›-5,πšŸπšŠπš•-1πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-3,πšŸπšŠπš•-2πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-1,πšŸπšŠπš•-5πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-1,πšŸπšŠπš•-7πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-0,πšŸπšŠπš•-9πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-1

The πšœπšŠπš–πšŽ_πšŠπ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’ constraint holds since:

  • The values 1, 9, 1, 5, 2, 1 assigned to πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 correspond to a permutation of the values 9, 1, 1, 1, 2, 5 assigned to πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.

  • The values 1, 2, 5, 7 and 6 are respectively used 3, 1, 1, 0 and 1 times.

Symmetries
  • Arguments are permutable w.r.t. permutation (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).

  • 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š…π™°π™»πš„π™΄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› or πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.πšŸπšŠπš› that does not belong to πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš• can be replaced by any other value that also does not belong to πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš•.

  • All occurrences of two distinct values in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.πšŸπšŠπš›, πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.πšŸπšŠπš› or πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš• can be swapped; all occurrences of a value in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.πšŸπšŠπš›, πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.πšŸπšŠπš› or πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš• can be renamed to any unused value.

Usage

See the πšœπšŠπš–πšŽ_πšŠπ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šžπš™ constraint.

Algorithm

The filtering algorithm presented inΒ [BeldiceanuKatrielThiel05b] can be reused for pruning the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 and the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 collection. This algorithm does not restrict the πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ variables of the πš…π™°π™»πš„π™΄πš‚ collection.

See also

implies: 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šœπšŠπš–πšŽ.

related: πš”_πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πšΒ (two overlapping πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš plus restriction on values).

specialisation: πšœπšŠπš–πšŽ_πšŠπ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šžπš™Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ replaced by πšπš’πš‘πšŽπš πš’πš—πšπšŽπš›πšŸπšŠπš•).

Keywords

application area: assignment.

combinatorial object: permutation, multiset.

constraint arguments: constraint between two collections of variables.

constraint type: value constraint.

filtering: flow.

modelling: equality between multisets.

problems: demand profile.

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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)
β€’ for all connected components: ππ’πŽπ”π‘π‚π„=ππ’πˆππŠ
β€’ ππ’πŽπ”π‘π‚π„=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|
β€’ ππ’πˆππŠ=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|

For all items of πš…π™°π™»πš„π™΄πš‚:

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1

Arc generator
π‘†πΈπΏπΉβ†¦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šœ)

Arc arity
Arc constraint(s)
πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ.πšŸπšŠπš›=πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš•
Graph property(ies)
𝐍𝐕𝐄𝐑𝐓𝐄𝐗=πš…π™°π™»πš„π™΄πš‚.πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.279.1 respectively show the initial and final graph associated with the first graph constraint of 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.

Figure 5.279.1. Initial and final graph of the πšœπšŠπš–πšŽ_πšŠπ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’ constraint
ctrs/same_and_global_cardinalityA
(a)
ctrs/same_and_global_cardinalityB
(b)