5.305. soft_same_interval_var

DESCRIPTIONLINKSGRAPH
Origin

Derived from πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•

Constraint

𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•_πšŸπšŠπš›(𝙲,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™»)

Synonym

𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•.

Arguments
π™²π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
𝙲β‰₯0
𝙲≀|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‚1|=|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

Let N i (respectively M i ) denote the number of variables of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 (respectively πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2) that take a value in the interval [πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»Β·i,πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»Β·i+πš‚π™Έπš‰π™΄_π™Έπ™½πšƒπ™΄πšπš…π™°π™»-1. 𝙲 is the minimum number of values to change in the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 and πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 collections so that for all integer i we have N i =M i .

Example
4,πšŸπšŠπš›-9,πšŸπšŠπš›-9,πšŸπšŠπš›-9,πšŸπšŠπš›-9,πšŸπšŠπš›-9,πšŸπšŠπš›-1,πšŸπšŠπš›-9,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-8,3

In the example, the fourth 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. Consequently the values of the collections 〈9,9,9,9,9,1βŒͺ and 〈9,1,1,1,1,8βŒͺ are respectively located within intervals [9,11], [9,11], [9,11], [9,11], [9,11], [0,2] and intervals [9,11], [0,2], [0,2], [0,2], [0,2], [6,8]. Since there is a correspondence between two pairs of intervals we must unset at least 6-2 items (6 is the number of items of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 and πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 collections). Consequently, the 𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•_πšŸπšŠπš› constraint holds since its first argument 𝙲 is set to 6-2.

Symmetries
  • Arguments are permutable w.r.t. permutation (𝙲) (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.

Usage

A soft πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš• constraint.

Algorithm

See algorithm of the 𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πšŸπšŠπš› constraint.

See also

hard version: πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•.

implies: 𝚜𝚘𝚏𝚝_𝚞𝚜𝚎𝚍_πš‹πš’_πš’πš—πšπšŽπš›πšŸπšŠπš•_πšŸπšŠπš›.

Keywords

constraint arguments: constraint between two collections of variables.

constraint type: soft constraint, relaxation, variable-based violation measure.

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)
ππ’πˆππŠ_ππ’πŽπ”π‘π‚π„=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|-𝙲

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.305.1 respectively show the initial and final graph associated with the Example slot. Since we use the ππ’πˆππŠ_ππ’πŽπ”π‘π‚π„ graph property, the source and sink vertices of the final graph are stressed with a double circle. The 𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•_πšŸπšŠπš› constraint holds since the cost 4 corresponds to the difference between the number of variables of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 and the sum over the different connected components of the minimum number of sources and sinks.

Figure 5.305.1. Initial and final graph of the 𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•_πšŸπšŠπš› constraint
ctrs/soft_same_interval_varA
(a)
ctrs/soft_same_interval_varB
(b)