5.140. global_cardinality_low_up_no_loop

DESCRIPTIONLINKSGRAPH
Origin

Derived from 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™ and πšπš›πšŽπšŽ.

Constraint

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™_πš—πš˜_πš•πš˜πš˜πš™π™Όπ™Έπ™½π™»π™Ύπ™Ύπ™Ώ,π™Όπ™°πš‡π™»π™Ύπ™Ύπ™Ώ,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πš…π™°π™»πš„π™΄πš‚

Synonym

𝚐𝚌𝚌_πš•πš˜πš _πšžπš™_πš—πš˜_πš•πš˜πš˜πš™.

Arguments
π™Όπ™Έπ™½π™»π™Ύπ™Ύπ™Ώπš’πš—πš
π™Όπ™°πš‡π™»π™Ύπ™Ύπ™Ώπš’πš—πš
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,πš˜πš–πš’πš—-πš’πš—πš,πš˜πš–πšŠπš‘-πš’πš—πš)
Restrictions
𝙼𝙸𝙽𝙻𝙾𝙾𝙿β‰₯0
π™Όπ™Έπ™½π™»π™Ύπ™Ύπ™Ώβ‰€π™Όπ™°πš‡π™»π™Ύπ™Ύπ™Ώ
π™Όπ™°πš‡π™»π™Ύπ™Ύπ™Ώβ‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄π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‚,πšŸπšŠπš›)
|πš…π™°π™»πš„π™΄πš‚|>0
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š’πš—,πš˜πš–π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™»πš„π™΄πš‚.πš˜πš–πš’πš—β‰₯0
πš…π™°π™»πš„π™΄πš‚.πš˜πš–πšŠπš‘β‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|
πš…π™°π™»πš„π™΄πš‚.πš˜πš–πš’πš—β‰€πš…π™°π™»πš„π™΄πš‚.πš˜πš–πšŠπš‘
Purpose

πš…π™°π™»πš„π™΄πš‚[i].πš˜πš–πš’πš— (1≀i≀|πš…π™°π™»πš„π™΄πš‚|) is less than or equal to the number of variables πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[j].πšŸπšŠπš› (jβ‰ i,1≀j≀|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|) that are assigned value πš…π™°π™»πš„π™΄πš‚[i].πšŸπšŠπš•.

πš…π™°π™»πš„π™΄πš‚[i].πš˜πš–πšŠπš‘ (1≀i≀|πš…π™°π™»πš„π™΄πš‚|) is greater than or equal to the number of variables πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[j].πšŸπšŠπš› (jβ‰ i,1≀j≀|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|) that are assigned value πš…π™°π™»πš„π™΄πš‚[i].πšŸπšŠπš•.

The number of assignments of the form πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›=i (i∈[1,|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|]) is greater than or equal to 𝙼𝙸𝙽𝙻𝙾𝙾𝙿 and less than or equal to π™Όπ™°πš‡π™»π™Ύπ™Ύπ™Ώ.

Example
1,1,1,1,8,6,πšŸπšŠπš•-1πš˜πš–πš’πš—-1πš˜πš–πšŠπš‘-1,πšŸπšŠπš•-5πš˜πš–πš’πš—-0πš˜πš–πšŠπš‘-0,πšŸπšŠπš•-6πš˜πš–πš’πš—-1πš˜πš–πšŠπš‘-2

The 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™_πš—πš˜_πš•πš˜πš˜πš™ constraint holds since:

  • Values 1, 5 and 6 are respectively assigned to the set of variables {πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[2].πšŸπšŠπš›} (i.e.,Β πš˜πš–πš’πš—=1≀1β‰€πš˜πš–πšŠπš‘=1), {} (i.e.,Β πš˜πš–πš’πš—=0≀0β‰€πš˜πš–πšŠπš‘=0) and {πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[4].πšŸπšŠπš›} (i.e.,Β πš˜πš–πš’πš—=1≀1β‰€πš˜πš–πšŠπš‘=2). Note that, due to the definition of the constraint, the fact that πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[1].πšŸπšŠπš› is assigned to 1 is not counted.

  • In addition the number of assignments of the form πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›=i (i∈[1,4]) is greater than or equal to 𝙼𝙸𝙽𝙻𝙾𝙾𝙿=1 and less than or equal to π™Όπ™°πš‡π™»π™Ύπ™Ύπ™Ώ=1.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>1
πš›πšŠπš—πšπšŽ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš›)>1
|πš…π™°π™»πš„π™΄πš‚|>1
πš…π™°π™»πš„π™΄πš‚.πš˜πš–πš’πš—β‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|
πš…π™°π™»πš„π™΄πš‚.πš˜πš–πšŠπš‘>0
πš…π™°π™»πš„π™΄πš‚.πš˜πš–πšŠπš‘β‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>|πš…π™°π™»πš„π™΄πš‚|
Symmetries
  • Items of πš…π™°π™»πš„π™΄πš‚ are permutable.

  • πš…π™°π™»πš„π™΄πš‚.πš˜πš–πš’πš— can be decreased to any value β‰₯0.

  • πš…π™°π™»πš„π™΄πš‚.πš˜πš–πšŠπš‘ can be increased to any value ≀|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|.

Usage

Within the context of the πšπš›πšŽπšŽ constraint the 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™_πš—πš˜_πš•πš˜πš˜πš™ constraint allows to model a minimum and maximum degree constraint on each vertex of our trees.

Algorithm

The flow algorithm that handles the original 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’ constraintΒ [Regin96] can be adapted to the context of the 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™_πš—πš˜_πš•πš˜πš˜πš™ constraint. This is done by creating an extra value node representing the loops corresponding to the roots of the trees.

See also

generalisation: 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š˜πš˜πš™Β (πšπš’πš‘πšŽπš πš’πš—πšπšŽπš›πšŸπšŠπš• replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

implied by: 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™.

related: πšπš›πšŽπšŽΒ (graph partitioning by a set of trees with degree restrictions).

root concept: 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 (assignment of a πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ to its position is ignored).

Keywords

constraint type: value constraint.

filtering: flow.

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‚

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)
β€’ 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•
β€’ πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ.πš”πšŽπš’β‰ πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš•
Graph property(ies)
β€’ 𝐍𝐕𝐄𝐑𝐓𝐄𝐗β‰₯πš…π™°π™»πš„π™΄πš‚.πš˜πš–πš’πš—
β€’ ππ•π„π‘π“π„π—β‰€πš…π™°π™»πš„π™΄πš‚.πš˜πš–πšŠπš‘

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

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šŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ.πš”πšŽπš’
Graph property(ies)
β€’ 𝐍𝐀𝐑𝐂β‰₯𝙼𝙸𝙽𝙻𝙾𝙾𝙿
β€’ ππ€π‘π‚β‰€π™Όπ™°πš‡π™»π™Ύπ™Ύπ™Ώ

Graph model

Since, within the context of the first graph constraint, we want to express one unary constraint for each value we use the β€œFor all items of πš…π™°π™»πš„π™΄πš‚β€ iterator. PartΒ (A) of FigureΒ 5.140.1 shows the initial graphs associated with each value 1, 5 and 6 of the πš…π™°π™»πš„π™΄πš‚ collection of the Example slot. PartΒ (B) of FigureΒ 5.140.1 shows the two corresponding final graphs respectively associated with values 1 and 6 that are both assigned to the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection (since value 5 is not assigned to any variable of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection the final graph associated with value 5 is empty). Since we use the 𝐍𝐕𝐄𝐑𝐓𝐄𝐗 graph property, the vertices of the final graphs are stressed in bold.

Figure 5.140.1. Initial and final graph of the 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™_πš—πš˜_πš•πš˜πš˜πš™ constraint
ctrs/global_cardinality_low_up_no_loopActrs/global_cardinality_low_up_no_loopB
(a) (b)