5.178. k_same

DESCRIPTIONLINKSGRAPH
Origin

[ElbassioniKatrielKutzMahajan05]

Constraint

πš”_πšœπšŠπš–πšŽ(πš‚π™΄πšƒπš‚)

Type
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›)
Argument
πš‚π™΄πšƒπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚜𝚎𝚝-πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚)
Restrictions
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšŸπšŠπš›)
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>0
πš›πšŽπššπšžπš’πš›πšŽπš(πš‚π™΄πšƒπš‚,𝚜𝚎𝚝)
|πš‚π™΄πšƒπš‚|>1
πšœπšŠπš–πšŽ_πšœπš’πš£πšŽ(πš‚π™΄πšƒπš‚,𝚜𝚎𝚝)
Purpose

Given |πš‚π™΄πšƒπš‚| sets, each containing the same number of domain variables, the πš”_πšœπšŠπš–πšŽ constraint enforces that the multisets of values assigned to each set are all identical.

Example
𝚜𝚎𝚝-πšŸπšŠπš›-1,πšŸπšŠπš›-9,πšŸπšŠπš›-1,πšŸπšŠπš›-5,πšŸπšŠπš›-2,πšŸπšŠπš›-1,𝚜𝚎𝚝-πšŸπšŠπš›-9,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-2,πšŸπšŠπš›-5,𝚜𝚎𝚝-πšŸπšŠπš›-5,πšŸπšŠπš›-2,πšŸπšŠπš›-1,πšŸπšŠπš›-1,πšŸπšŠπš›-9,πšŸπšŠπš›-1

The πš”_πšœπšŠπš–πšŽ constraint holds since:

  • The first and second collections of variables are assigned to the same multiset.

  • The second and third collections of variables are also assigned to the same multiset.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>1
Symmetries
  • Items of πš‚π™΄πšƒπš‚ are permutable.

  • Items of πš‚π™΄πšƒπš‚.𝚜𝚎𝚝 are permutable.

  • All occurrences of two distinct values of πš‚π™΄πšƒπš‚.𝚜𝚎𝚝.πšŸπšŠπš› can be swapped; all occurrences of a value of πš‚π™΄πšƒπš‚.𝚜𝚎𝚝.πšŸπšŠπš› can be renamed to any unused value.

Remark

It was shown inΒ [ElbassioniKatrielKutzMahajan05] that, finding out whether the πš”_πšœπšŠπš–πšŽ constraint has a solution or not is NP-hard when we have more than one πšœπšŠπš–πšŽ constraint. This was achieved by reduction from 3-dimensional-matching in the context where we have 2 πšœπšŠπš–πšŽ constraints.

See also

common keyword: πš”_πšœπšŠπš–πšŽ_πš’πš—πšπšŽπš›πšŸπšŠπš•, πš”_πšœπšŠπš–πšŽ_πš–πš˜πšπšžπš•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—Β (system of constraints).

implies: πš”_𝚞𝚜𝚎𝚍_πš‹πš’.

part of system of constraints: πšœπšŠπš–πšŽ.

used in graph description: πšœπšŠπš–πšŽ.

Keywords

combinatorial object: permutation, multiset.

complexity: 3-dimensional-matching.

constraint type: system of constraints, decomposition.

modelling: equality between multisets.

Arc input(s)

πš‚π™΄πšƒπš‚

Arc generator
π‘ƒπ΄π‘‡π»β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚜𝚎𝚝1,𝚜𝚎𝚝2)

Arc arity
Arc constraint(s)
πšœπšŠπš–πšŽ(𝚜𝚎𝚝1.𝚜𝚎𝚝,𝚜𝚎𝚝2.𝚜𝚎𝚝)
Graph property(ies)
𝐍𝐀𝐑𝐂=|πš‚π™΄πšƒπš‚|-1

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.178.1 respectively show the initial and final graph associated with the Example slot. To each vertex corresponds a collection of variables, while to each arc corresponds a πšœπšŠπš–πšŽ constraint.

Figure 5.178.1. Initial and final graph of the πš”_πšœπšŠπš–πšŽ constraint
ctrs/k_sameActrs/k_sameB
(a) (b)