5.141. global_cardinality_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’ 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š˜πš˜πš™(𝙽𝙻𝙾𝙾𝙿,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š˜πš˜πš™.

Arguments
π™½π™»π™Ύπ™Ύπ™ΏπšπšŸπšŠπ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šŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-πšπšŸπšŠπš›)
Restrictions
𝙽𝙻𝙾𝙾𝙿β‰₯0
𝙽𝙻𝙾𝙾𝙿≀|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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‚|
Purpose

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 (1≀i≀|πš…π™°π™»πš„π™΄πš‚|) is 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 equal to 𝙽𝙻𝙾𝙾𝙿.

Example
1,1,1,8,6,πšŸπšŠπš•-1π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-0,πšŸπšŠπš•-6πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-1

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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 1 occurrence of value 1), {} (i.e.,Β no occurrence of value 5) and {πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[4].πšŸπšŠπš›} (i.e.,Β 1 occurrence of value 6). 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 equal to 𝙽𝙻𝙾𝙾𝙿=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šŠπš—πšπšŽ(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)>1
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>|πš…π™°π™»πš„π™΄πš‚|
Symmetry

Items of πš…π™°π™»πš„π™΄πš‚ are permutable.

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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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

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

specialisation: 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™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

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)
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

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)
𝐍𝐀𝐑𝐂=𝙽𝙻𝙾𝙾𝙿

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 5.141.1 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 5.141.1 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 5.141.1. 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š˜πš˜πš™ constraint
ctrs/global_cardinality_no_loopActrs/global_cardinality_no_loopB
(a) (b)