5.62. common

DESCRIPTIONLINKSGRAPH
Origin

N.Β Beldiceanu

Constraint

πšŒπš˜πš–πš–πš˜πš—(𝙽𝙲𝙾𝙼𝙼𝙾𝙽1,𝙽𝙲𝙾𝙼𝙼𝙾𝙽2,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)

Arguments
𝙽𝙲𝙾𝙼𝙼𝙾𝙽1πšπšŸπšŠπš›
𝙽𝙲𝙾𝙼𝙼𝙾𝙽2πšπšŸπšŠπš›
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›)
Restrictions
𝙽𝙲𝙾𝙼𝙼𝙾𝙽1β‰₯0
𝙽𝙲𝙾𝙼𝙼𝙾𝙽1≀|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|
𝙽𝙲𝙾𝙼𝙼𝙾𝙽2β‰₯0
𝙽𝙲𝙾𝙼𝙼𝙾𝙽2≀|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›)
Purpose

𝙽𝙲𝙾𝙼𝙼𝙾𝙽1 is the number of variables of the collection of variables πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 taking a value in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.

𝙽𝙲𝙾𝙼𝙼𝙾𝙽2 is the number of variables of the collection of variables πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 taking a value in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.

Example
3,4,1,9,1,5,πšŸπšŠπš›-2,πšŸπšŠπš›-1,πšŸπšŠπš›-9,πšŸπšŠπš›-9,πšŸπšŠπš›-6,πšŸπšŠπš›-9

The πšŒπš˜πš–πš–πš˜πš— constraint holds since:

  • Its first argument 𝙽𝙲𝙾𝙼𝙼𝙾𝙽1=3 corresponds to the number of values of the collection 〈1,9,1,5βŒͺ that occur within 〈2,1,9,9,6,9βŒͺ.

  • Its second argument 𝙽𝙲𝙾𝙼𝙼𝙾𝙽2=4 corresponds to the number of values of the collection 〈2,1,9,9,6,9βŒͺ that occur within 〈1,9,1,5βŒͺ.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|>1
πš›πšŠπš—πšπšŽ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.πšŸπšŠπš›)>1
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|>1
πš›πšŠπš—πšπšŽ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.πšŸπšŠπš›)>1
Symmetries
  • Arguments are permutable w.r.t. permutation (𝙽𝙲𝙾𝙼𝙼𝙾𝙽1,𝙽𝙲𝙾𝙼𝙼𝙾𝙽2) (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.

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

Remark

It was shown inΒ [BessiereHebrardHnichWalsh04a] that, finding out whether the πšŒπš˜πš–πš–πš˜πš— constraint has a solution or not is NP-hard. This was achieved by reduction from 3-SAT.

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š—_πš’πš—πšπšŽπš›πšœπšŽπšŒπšπš’πš˜πš—, πš—πšŸπšŠπš•πšžπšŽ_πš˜πš—_πš’πš—πšπšŽπš›πšœπšŽπšŒπšπš’πš˜πš—, πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšœπšŽπšŒπšπš’πš˜πš—Β (constraint on the intersection).

generalisation: πšŒπš˜πš–πš–πš˜πš—_πš’πš—πšπšŽπš›πšŸπšŠπš•Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ/πšŒπš˜πš—πšœπšπšŠπš—πš), πšŒπš˜πš–πš–πš˜πš—_πš–πš˜πšπšžπš•πš˜Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ mod πšŒπš˜πš—πšœπšπšŠπ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—Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ replaced by 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—).

related: πšŠπš–πš˜πš—πš_πšŸπšŠπš›, πš›πš˜πš˜πšπšœ.

root concept: πšŠπš–πš˜πš—πš.

specialisation: 𝚞𝚜𝚎𝚜 (𝙽𝙲𝙾𝙼𝙼𝙾𝙽2=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|).

Keywords

complexity: 3-SAT.

constraint arguments: constraint between two collections of variables.

constraint type: constraint on the intersection.

final graph structure: acyclic, bipartite, no loop.

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€’ ππ’πŽπ”π‘π‚π„=𝙽𝙲𝙾𝙼𝙼𝙾𝙽1
β€’ ππ’πˆππŠ=𝙽𝙲𝙾𝙼𝙼𝙾𝙽2

Graph class
β€’ π™°π™²πšˆπ™²π™»π™Έπ™²
β€’ π™±π™Έπ™Ώπ™°πšπšƒπ™Έπšƒπ™΄
β€’ 𝙽𝙾_𝙻𝙾𝙾𝙿

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.62.1 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 the final graph has only 3 sources and 4 sinks the variables 𝙽𝙲𝙾𝙼𝙼𝙾𝙽1 and 𝙽𝙲𝙾𝙼𝙼𝙾𝙽2 are respectively equal to 3 and 4. Note that all the vertices corresponding to the variables that take values 5, 2 or 6 were removed from the final graph since there is no arc for which the associated equality constraint holds.

Figure 5.62.1. Initial and final graph of the πšŒπš˜πš–πš–πš˜πš— constraint
ctrs/commonActrs/commonB
(a) (b)