5.59. colored_matrix

DESCRIPTIONLINKS
Origin

KOALOG

Constraint

πšŒπš˜πš•πš˜πš›πšŽπš_πš–πšŠπšπš›πš’πš‘(𝙲,𝙻,𝙺,π™Όπ™°πšƒπšπ™Έπš‡,π™²π™Ώπšπ™Ύπ™Ή,π™»π™Ώπšπ™Ύπ™Ή)

Synonyms

πšŒπš˜πš•πš˜πšžπš›πšŽπš_πš–πšŠπšπš›πš’πš‘, πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’_πš–πšŠπšπš›πš’πš‘, πšŒπšŠπš›πš_πš–πšŠπšπš›πš’πš‘.

Arguments
π™²πš’πš—πš
π™»πš’πš—πš
π™Ίπš’πš—πš
π™Όπ™°πšƒπšπ™Έπš‡πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŒπš˜πš•πšžπš–πš—-πš’πš—πš,πš•πš’πš—πšŽ-πš’πš—πš,πšŸπšŠπš›-πšπšŸπšŠπš›)
π™²π™Ώπšπ™Ύπ™ΉπšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŒπš˜πš•πšžπš–πš—-πš’πš—πš,πšŸπšŠπš•-πš’πš—πš,πš—πš˜πšŒπšŒ-πšπšŸπšŠπš›)
π™»π™Ώπšπ™Ύπ™ΉπšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš•πš’πš—πšŽ-πš’πš—πš,πšŸπšŠπš•-πš’πš—πš,πš—πš˜πšŒπšŒ-πšπšŸπšŠπš›)
Restrictions
𝙲β‰₯0
𝙻β‰₯0
𝙺β‰₯0
πš›πšŽπššπšžπš’πš›πšŽπš(π™Όπ™°πšƒπšπ™Έπš‡,[πšŒπš˜πš•πšžπš–πš—,πš•πš’πš—πšŽ,πšŸπšŠπš›])
πš’πš—πšŒπš›πšŽπšŠπšœπš’πš—πš_𝚜𝚎𝚚(π™Όπ™°πšƒπšπ™Έπš‡,[πšŒπš˜πš•πšžπš–πš—,πš•πš’πš—πšŽ])
|π™Όπ™°πšƒπšπ™Έπš‡|=𝙲*𝙻+𝙲+𝙻+1
π™Όπ™°πšƒπšπ™Έπš‡.πšŒπš˜πš•πšžπš–πš—β‰₯0
π™Όπ™°πšƒπšπ™Έπš‡.πšŒπš˜πš•πšžπš–πš—β‰€π™²
π™Όπ™°πšƒπšπ™Έπš‡.πš•πš’πš—πšŽβ‰₯0
π™Όπ™°πšƒπšπ™Έπš‡.πš•πš’πš—πšŽβ‰€π™»
π™Όπ™°πšƒπšπ™Έπš‡.πšŸπšŠπš›β‰₯0
π™Όπ™°πšƒπšπ™Έπš‡.πšŸπšŠπš›β‰€π™Ί
πš›πšŽπššπšžπš’πš›πšŽπš(π™²π™Ώπšπ™Ύπ™Ή,[πšŒπš˜πš•πšžπš–πš—,πšŸπšŠπš•,πš—πš˜πšŒπšŒ])
πš’πš—πšŒπš›πšŽπšŠπšœπš’πš—πš_𝚜𝚎𝚚(π™²π™Ώπšπ™Ύπ™Ή,[πšŒπš˜πš•πšžπš–πš—,πšŸπšŠπš•])
|π™²π™Ώπšπ™Ύπ™Ή|=𝙲*𝙺+𝙲+𝙺+1
π™²π™Ώπšπ™Ύπ™Ή.πšŒπš˜πš•πšžπš–πš—β‰₯0
π™²π™Ώπšπ™Ύπ™Ή.πšŒπš˜πš•πšžπš–πš—β‰€π™²
π™²π™Ώπšπ™Ύπ™Ή.πšŸπšŠπš•β‰₯0
π™²π™Ώπšπ™Ύπ™Ή.πšŸπšŠπš•β‰€π™Ί
πš›πšŽπššπšžπš’πš›πšŽπš(π™»π™Ώπšπ™Ύπ™Ή,[πš•πš’πš—πšŽ,πšŸπšŠπš•,πš—πš˜πšŒπšŒ])
πš’πš—πšŒπš›πšŽπšŠπšœπš’πš—πš_𝚜𝚎𝚚(π™»π™Ώπšπ™Ύπ™Ή,[πš•πš’πš—πšŽ,πšŸπšŠπš•])
|π™»π™Ώπšπ™Ύπ™Ή|=𝙻*𝙺+𝙻+𝙺+1
π™»π™Ώπšπ™Ύπ™Ή.πš•πš’πš—πšŽβ‰₯0
π™»π™Ώπšπ™Ύπ™Ή.πš•πš’πš—πšŽβ‰€π™»
π™»π™Ώπšπ™Ύπ™Ή.πšŸπšŠπš•β‰₯0
π™»π™Ώπšπ™Ύπ™Ή.πšŸπšŠπš•β‰€π™Ί
Purpose

Given a matrix of domain variables, imposes a πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’ constraint involving cardinality variables on each column and each row of the matrix.

Example
1,2,4,πšŒπš˜πš•πšžπš–πš—-0πš•πš’πš—πšŽ-0πšŸπšŠπš›-3,πšŒπš˜πš•πšžπš–πš—-0πš•πš’πš—πšŽ-1πšŸπšŠπš›-1,πšŒπš˜πš•πšžπš–πš—-0πš•πš’πš—πšŽ-2πšŸπšŠπš›-3,πšŒπš˜πš•πšžπš–πš—-1πš•πš’πš—πšŽ-0πšŸπšŠπš›-4,πšŒπš˜πš•πšžπš–πš—-1πš•πš’πš—πšŽ-1πšŸπšŠπš›-4,πšŒπš˜πš•πšžπš–πš—-1πš•πš’πš—πšŽ-2πšŸπšŠπš›-3,πšŒπš˜πš•πšžπš–πš—-0πšŸπšŠπš•-0πš—πš˜πšŒπšŒ-0,πšŒπš˜πš•πšžπš–πš—-0πšŸπšŠπš•-1πš—πš˜πšŒπšŒ-1,πšŒπš˜πš•πšžπš–πš—-0πšŸπšŠπš•-2πš—πš˜πšŒπšŒ-0,πšŒπš˜πš•πšžπš–πš—-0πšŸπšŠπš•-3πš—πš˜πšŒπšŒ-2,πšŒπš˜πš•πšžπš–πš—-0πšŸπšŠπš•-4πš—πš˜πšŒπšŒ-0,πšŒπš˜πš•πšžπš–πš—-1πšŸπšŠπš•-0πš—πš˜πšŒπšŒ-0,πšŒπš˜πš•πšžπš–πš—-1πšŸπšŠπš•-1πš—πš˜πšŒπšŒ-0,πšŒπš˜πš•πšžπš–πš—-1πšŸπšŠπš•-2πš—πš˜πšŒπšŒ-0,πšŒπš˜πš•πšžπš–πš—-1πšŸπšŠπš•-3πš—πš˜πšŒπšŒ-1,πšŒπš˜πš•πšžπš–πš—-1πšŸπšŠπš•-4πš—πš˜πšŒπšŒ-2,πš•πš’πš—πšŽ-0πšŸπšŠπš•-0πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-0πšŸπšŠπš•-1πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-0πšŸπšŠπš•-2πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-0πšŸπšŠπš•-3πš—πš˜πšŒπšŒ-1,πš•πš’πš—πšŽ-0πšŸπšŠπš•-4πš—πš˜πšŒπšŒ-1,πš•πš’πš—πšŽ-1πšŸπšŠπš•-0πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-1πšŸπšŠπš•-1πš—πš˜πšŒπšŒ-1,πš•πš’πš—πšŽ-1πšŸπšŠπš•-2πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-1πšŸπšŠπš•-3πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-1πšŸπšŠπš•-4πš—πš˜πšŒπšŒ-1,πš•πš’πš—πšŽ-2πšŸπšŠπš•-0πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-2πšŸπšŠπš•-1πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-2πšŸπšŠπš•-2πš—πš˜πšŒπšŒ-0,πš•πš’πš—πšŽ-2πšŸπšŠπš•-3πš—πš˜πšŒπšŒ-2,πš•πš’πš—πšŽ-2πšŸπšŠπš•-4πš—πš˜πšŒπšŒ-0
Typical
𝙲β‰₯1
𝙻β‰₯1
𝙺β‰₯1
πš›πšŠπš—πšπšŽ(π™Όπ™°πšƒπšπ™Έπš‡.πšŸπšŠπš›)>1
Remark

WithinΒ [ReginGomes04] the πšŒπš˜πš•πš˜πš›πšŽπš_πš–πšŠπšπš›πš’πš‘ constraint is called πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’_πš–πšŠπšπš›πš’πš‘.

Algorithm

The filtering algorithm described inΒ [ReginGomes04] is based on network flow and does not achieve arc-consistency in general. However, when the number of values is restricted to two, the algorithmΒ [ReginGomes04] achieves arc-consistency on the variables of the matrix. This corresponds in fact to a generalisation of the problem called "Matrices composed of 0's and 1's" presented by Ford and FulkersonΒ [FordFulkerson62].

See also

common keyword: πš”_πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πšΒ (system of constraints).

part of system of constraints: πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’.

related to a common problem: πšœπšŠπš–πšŽΒ (matrix reconstruction problem).

Keywords

constraint type: system of constraints, predefined constraint, timetabling constraint.

modelling: matrix, matrix model.