5.308. soft_same_var

DESCRIPTIONLINKSGRAPH
Origin

[vanHoeve05]

Constraint

𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_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)

Synonym

𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ.

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›)
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›)
Purpose

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 the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 collection correspond to the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 collection according to a permutation.

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

As illustrated by FigureΒ 5.308.1, there is a correspondence between two pairs of values of the collections 〈9,9,9,9,9,1βŒͺ and 〈9,1,1,1,1,8βŒͺ. Consequently, 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). The 𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πšŸπšŠπš› constraint holds since its first argument 𝙲 is set to 6-2.

Figure 5.308.1. Correspondence between collection 〈9,9,9,9,9,1βŒͺ and collection 〈9,1,1,1,1,8βŒͺ
ctrs/soft_same_var1
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).

  • 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.

Usage

A soft πšœπšŠπš–πšŽ constraint.

Algorithm

[vanHoeve05].

See also

hard version: πšœπšŠπš–πšŽ.

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

Keywords

constraint arguments: constraint between two collections of variables.

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

filtering: minimum cost flow.

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)
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 5.308.2 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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 5.308.2. Initial and final graph of the 𝚜𝚘𝚏𝚝_πšœπšŠπš–πšŽ_πšŸπšŠπš› constraint
ctrs/soft_same_varA
(a)
ctrs/soft_same_varB
(b)