5.157. in_same_partition

DESCRIPTIONLINKSGRAPHAUTOMATON
Origin

Used for defining several entries of this catalog.

Constraint

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š…π™°πš1,πš…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1πšπšŸπšŠπš›
πš…π™°πš2π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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

Enforce πš…π™°πš1 and πš…π™°πš2 to be respectively assigned to values v 1 and v 2 that both belong to a same partition of the collection π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

Example
6,2,πš™-1,3,πš™-4,πš™-2,6

The 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— constraint holds since its first and second arguments πš…π™°πš1=6 and πš…π™°πš2=2 both belong to the third partition 〈2,6βŒͺ of its third argument π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

Typical
πš…π™°πš1β‰ πš…π™°πš2
Symmetries
  • Arguments are permutable w.r.t. permutation (πš…π™°πš1,πš…π™°πš2) (π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚).

  • 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.

Used in

πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_πš™πšŠπš›πšπš’πšπš’πš˜πš—, πš‹πšŠπš•πšŠπš—πšŒπšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—, πšŒπš‘πšŠπš—πšπšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—, πšŒπš˜πš–π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šŒπš•πšŠπšœπšœ, πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—, 𝚜𝚘𝚏𝚝_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›, 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›, 𝚞𝚜𝚎𝚍_πš‹πš’_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

See also

common keyword: πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_πš™πšŠπš›πšπš’πšπš’πš˜πš—Β (partition), πš’πš—Β (value constraint).

used in graph description: πš’πš—.

Keywords

characteristic of a constraint: partition, automaton, automaton without counters, reified automaton constraint, derived collection.

constraint arguments: binary constraint.

constraint network structure: centered cyclic(2) constraint network(1).

constraint type: value constraint.

filtering: arc-consistency.

Derived Collection
πšŒπ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šπšŽπš–(πšŸπšŠπš›-πš…π™°πš1),πš’πšπšŽπš–(πšŸπšŠπš›-πš…π™°πš2)]
Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚

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šœ,πš™πšŠπš›πšπš’πšπš’πš˜πš—πšœ)

Arc arity
Arc constraint(s)
πš’πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ.πšŸπšŠπš›,πš™πšŠπš›πšπš’πšπš’πš˜πš—πšœ.πš™)
Graph property(ies)
β€’ ππ’πŽπ”π‘π‚π„=2
β€’ ππ’πˆππŠ=1

Graph model

πš…π™°πš1 and πš…π™°πš2 are put together in the derived collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚. Since both πš…π™°πš1 and πš…π™°πš2 should take their value in one of the partition depicted by the π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ collection, the final graph should have two sources corresponding respectively to πš…π™°πš1 and πš…π™°πš2. Since two, possibly distinct, values should be assigned to πš…π™°πš1 and πš…π™°πš2 and since these values belong to the same partition p the final graph should only have one sink. This sink corresponds in fact to partition p.

PartsΒ (A) andΒ (B) of FigureΒ 5.157.1 respectively show the initial and final graph associated with the Example slot. Since we both 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 shown with a double circle.

Figure 5.157.1. Initial and final graph of the 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— constraint
ctrs/in_same_partitionActrs/in_same_partitionB
(a) (b)
Signature

Note that the sinks of the initial graph cannot become sources of the final graph since isolated vertices are eliminated from the final graph. Since the final graph contains two sources it also includes one arc between a source and a sink. Therefore the minimum number of sinks of the final graph is equal to one. So we can rewrite ππ’πˆππŠ=1 to ππ’πˆππŠβ‰₯1 and simplify ππ’πˆππŠ Β― Μ² to ππ’πˆππŠ Β―.

Automaton

FigureΒ 5.157.2 depicts the automaton associated with the 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— constraint. Let πš…π™°π™»πš„π™΄πš‚ i be the πš™ attribute of the i th item of the π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ collection. To each triple (πš…π™°πš1,πš…π™°πš2,πš…π™°π™»πš„π™΄πš‚ i ) corresponds a 0-1 signature variable πš‚ i as well as the following signature constraint: ((πš…π™°πš1βˆˆπš…π™°π™»πš„π™΄πš‚ i )∧(πš…π™°πš2βˆˆπš…π™°π™»πš„π™΄πš‚ i ))β‡”πš‚ i .

Figure 5.157.2. Automaton of the 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— constraint
ctrs/in_same_partition1
Figure 5.157.3. Hypergraph of the reformulation corresponding to the automaton of the 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— constraint
ctrs/in_same_partition2