5.63. common_interval

DESCRIPTIONLINKSGRAPH
Origin

Derived from πšŒπš˜πš–πš–πš˜πš—.

Constraint

πšŒπš˜πš–πš–πš˜πš—_πš’πš—πšπšŽπš›πšŸπšŠπš•π™½π™²π™Ύπ™Όπ™Όπ™Ύπ™½1,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,πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»

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›)
πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»πš’πš—π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›)
πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»>0
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 one of the intervals derived from the values assigned to the variables of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2: To each value v assigned to a variable of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 we associate the interval [πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»Β·βŒŠv/πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»βŒ‹,πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»Β·βŒŠv/πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»βŒ‹+πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»-1].

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 one of the intervals derived from the values assigned to the variables of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1: To each value v assigned to a variable of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 we associate the interval [πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»Β·βŒŠv/πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»βŒ‹,πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»Β·βŒŠv/πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»βŒ‹+πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»-1].

Example
3,2,8,6,6,0,πšŸπšŠπš›-7,πšŸπšŠπš›-3,πšŸπšŠπš›-3,πšŸπšŠπš›-3,πšŸπšŠπš›-3,πšŸπšŠπš›-7,3

In the example, the last argument πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»=3 defines the following family of intervals [3Β·k,3Β·k+2], where k is an integer. As a consequence the items of collection 〈8,6,6,0βŒͺ respectively correspond to intervals [6,8], [6,8], [6,8] and [0,2]. Similarly the items of collection 〈7,3,3,3,3,7βŒͺ respectively correspond to intervals [6,8], [3,5], [3,5], [3,5], [3,5], [6,8]. The πšŒπš˜πš–πš–πš˜πš—_πš’πš—πšπšŽπš›πšŸπšŠπš• constraint holds since:

  • Its first argument 𝙽𝙲𝙾𝙼𝙼𝙾𝙽1=3 is the number of intervals associated with the items of collection 〈8,6,6,0βŒͺ that also correspond to intervals associated with 〈7,3,3,3,3,7βŒͺ.

  • Its second argument 𝙽𝙲𝙾𝙼𝙼𝙾𝙽2=2 is the number of intervals associated with the items of collection 〈7,3,3,3,3,7βŒͺ that also correspond to intervals associated with 〈8,6,6,0βŒͺ.

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
πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»>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) (πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»).

  • 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.

  • An occurrence of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.πšŸπšŠπš› that belongs to the k-th interval, of size πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™», can be replaced by any other value of the same interval.

  • An occurrence of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.πšŸπšŠπš› that belongs to the k-th interval, of size πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™», can be replaced by any other value of the same interval.

See also

specialisation: πšŒπš˜πš–πš–πš˜πš—Β (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 replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

Keywords

constraint arguments: constraint between two collections of variables.

final graph structure: acyclic, bipartite, no loop.

modelling: interval.

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š‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»=πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›/πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»
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 5.63.1 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 graph has only 3 sources and 2 sinks the variables 𝙽𝙲𝙾𝙼𝙼𝙾𝙽1 and 𝙽𝙲𝙾𝙼𝙼𝙾𝙽2 are respectively equal to 3 and 2. Note that the vertices corresponding to the variables that take values 0 or 3 were removed from the final graph since there is no arc for which the associated arc constraint holds.

Figure 5.63.1. Initial and final graph of the πšŒπš˜πš–πš–πš˜πš—_πš’πš—πšπšŽπš›πšŸπšŠπš• constraint
ctrs/common_intervalActrs/common_intervalB
(a) (b)