5.90. cycle_card_on_path

DESCRIPTIONLINKSGRAPH
Origin

CHIP

Constraint

πšŒπš’πšŒπš•πšŽ_πšŒπšŠπš›πš_πš˜πš—_πš™πšŠπšπš‘(π™½π™²πšˆπ™²π™»π™΄,π™½π™Ύπ™³π™΄πš‚,π™°πšƒπ™»π™΄π™°πš‚πšƒ,π™°πšƒπ™Όπ™Ύπš‚πšƒ,π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽,πš…π™°π™»πš„π™΄πš‚)

Arguments
π™½π™²πšˆπ™²π™»π™΄πšπšŸπšŠπš›
π™½π™Ύπ™³π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš’πš—πšπšŽπš‘-πš’πš—πš,𝚜𝚞𝚌𝚌-πšπšŸπšŠπš›,πšŒπš˜πš•πš˜πšžπš›-πšπšŸπšŠπš›)
π™°πšƒπ™»π™΄π™°πš‚πšƒπš’πš—πš
π™°πšƒπ™Όπ™Ύπš‚πšƒπš’πš—πš
π™Ώπ™°πšƒπ™·_π™»π™΄π™½πš’πš—πš
πš…π™°π™»πš„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š—πš)
Restrictions
π™½π™²πšˆπ™²π™»π™΄β‰₯1
π™½π™²πšˆπ™²π™»π™΄β‰€|π™½π™Ύπ™³π™΄πš‚|
πš›πšŽπššπšžπš’πš›πšŽπš(π™½π™Ύπ™³π™΄πš‚,[πš’πš—πšπšŽπš‘,𝚜𝚞𝚌𝚌,πšŒπš˜πš•πš˜πšžπš›])
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰₯1
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰€|π™½π™Ύπ™³π™΄πš‚|
πšπš’πšœπšπš’πš—πšŒπš(π™½π™Ύπ™³π™΄πš‚,πš’πš—πšπšŽπš‘)
π™½π™Ύπ™³π™΄πš‚.𝚜𝚞𝚌𝚌β‰₯1
π™½π™Ύπ™³π™΄πš‚.πšœπšžπšŒπšŒβ‰€|π™½π™Ύπ™³π™΄πš‚|
π™°πšƒπ™»π™΄π™°πš‚πšƒβ‰₯0
π™°πšƒπ™»π™΄π™°πš‚πšƒβ‰€π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽
π™°πšƒπ™Όπ™Ύπš‚πšƒβ‰₯π™°πšƒπ™»π™΄π™°πš‚πšƒ
π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽β‰₯0
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
πšπš’πšœπšπš’πš—πšŒπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
Purpose

Consider a digraph G described by the π™½π™Ύπ™³π™΄πš‚ collection. π™½π™²πšˆπ™²π™»π™΄ is the number of circuits for covering G in such a way that each vertex belongs to one single circuit. In addition the following constraint must also hold: on each set of π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽 consecutive distinct vertices of each final circuit, the number of vertices for which the attribute colour takes his value in the collection of values πš…π™°π™»πš„π™΄πš‚ should be located within the range [π™°πšƒπ™»π™΄π™°πš‚πšƒ,π™°πšƒπ™Όπ™Ύπš‚πšƒ].

Example
2,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-7πšŒπš˜πš•πš˜πšžπš›-2,πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-4πšŒπš˜πš•πš˜πšžπš›-3,πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-8πšŒπš˜πš•πš˜πšžπš›-2,πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-9πšŒπš˜πš•πš˜πšžπš›-1,πš’πš—πšπšŽπš‘-5𝚜𝚞𝚌𝚌-1πšŒπš˜πš•πš˜πšžπš›-2,πš’πš—πšπšŽπš‘-6𝚜𝚞𝚌𝚌-2πšŒπš˜πš•πš˜πšžπš›-1,πš’πš—πšπšŽπš‘-7𝚜𝚞𝚌𝚌-5πšŒπš˜πš•πš˜πšžπš›-1,πš’πš—πšπšŽπš‘-8𝚜𝚞𝚌𝚌-6πšŒπš˜πš•πš˜πšžπš›-1,πš’πš—πšπšŽπš‘-9𝚜𝚞𝚌𝚌-3πšŒπš˜πš•πš˜πšžπš›-1,1,2,3,1

The constraint πšŒπš’πšŒπš•πšŽ_πšŒπšŠπš›πš_πš˜πš—_πš™πšŠπšπš‘ holds since the vertices of the π™½π™Ύπ™³π™΄πš‚ collection correspond to a set of disjoint circuits and since, for each set of 3 (i.e.,Β π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽=3) consecutive vertices, colour 1 (i.e.,Β the value provided by the πš…π™°π™»πš„π™΄πš‚ collection) occurs at least once (i.e.,Β π™°πšƒπ™»π™΄π™°πš‚πšƒ=1) and at most twice (i.e.,Β π™°πšƒπ™Όπ™Ύπš‚πšƒ=2).

Typical
|π™½π™Ύπ™³π™΄πš‚|>2
π™½π™²πšˆπ™²π™»π™΄<|π™½π™Ύπ™³π™΄πš‚|
π™°πšƒπ™»π™΄π™°πš‚πšƒ<π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽
π™°πšƒπ™Όπ™Ύπš‚πšƒ>0
π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽>1
|π™½π™Ύπ™³π™΄πš‚|>|πš…π™°π™»πš„π™΄πš‚|
Symmetries
  • Items of π™½π™Ύπ™³π™΄πš‚ are permutable.

  • An occurrence of a value of π™½π™Ύπ™³π™΄πš‚.πšŒπš˜πš•πš˜πšžπš› that belongs to πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš• (resp. does not belong to πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš•) can be replaced by any other value in πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš• (resp. not in πš…π™°π™»πš„π™΄πš‚.πšŸπšŠπš•).

  • π™°πšƒπ™»π™΄π™°πš‚πšƒ can be decreased to any value β‰₯0.

  • π™°πšƒπ™Όπ™Ύπš‚πšƒ can be increased.

  • Items of πš…π™°π™»πš„π™΄πš‚ are permutable.

Usage

Assume that the vertices of G are partitioned into the following two categories:

  • Clients to visit.

  • Depots where one can reload a vehicle.

Using the πšŒπš’πšŒπš•πšŽ_πšŒπšŠπš›πš_πš˜πš—_πš™πšŠπšπš‘ constraint we can express a constraint like: after visiting three consecutive clients we should visit a depot. This is typically not possible with the πšŠπšπš–πš˜πšœπš constraint since we do not know in advance the set of variables involved in the πšŠπšπš–πš˜πšœπš constraint.

Remark

This constraint is a special case of the πšœπšŽπššπšžπšŽπš—πšŒπšŽ parameter of the πšŒπš’πšŒπš•πšŽ constraint of CHIPΒ [Bourreau99].

See also

common keyword: πšŒπš’πšŒπš•πšŽΒ (graph partitioning constraint).

used in graph description: πšŠπš–πš˜πš—πš_πš•πš˜πš _πšžπš™.

Keywords

characteristic of a constraint: coloured.

combinatorial object: sequence.

constraint type: graph constraint, graph partitioning constraint, sliding sequence constraint.

final graph structure: connected component, one_succ.

Arc input(s)

π™½π™Ύπ™³π™΄πš‚

Arc generator
πΆπΏπΌπ‘„π‘ˆπΈβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš—πš˜πšπšŽπšœ1,πš—πš˜πšπšŽπšœ2)

Arc arity
Arc constraint(s)
πš—πš˜πšπšŽπšœ1.𝚜𝚞𝚌𝚌=πš—πš˜πšπšŽπšœ2.πš’πš—πšπšŽπš‘
Graph property(ies)
β€’ 𝐍𝐓𝐑𝐄𝐄=0
β€’ 𝐍𝐂𝐂=π™½π™²πšˆπ™²π™»π™΄

Graph class
𝙾𝙽𝙴_πš‚πš„π™²π™²

Sets
𝖯𝖠𝖳𝖧_𝖫𝖀𝖭𝖦𝖳𝖧(π™Ώπ™°πšƒπ™·_𝙻𝙴𝙽)β†¦πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ-πšŒπš˜πš•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›),πš’πšπšŽπš–(πšŸπšŠπš›-π™½π™Ύπ™³π™΄πš‚.πšŒπš˜πš•πš˜πšžπš›)]

Constraint(s) on sets
πšŠπš–πš˜πš—πš_πš•πš˜πš _πšžπš™(π™°πšƒπ™»π™΄π™°πš‚πšƒ,π™°πšƒπ™Όπ™Ύπš‚πšƒ,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ,πš…π™°π™»πš„π™΄πš‚)
Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.90.1 respectively show the initial and final graph associated with the Example slot. Since we use the 𝐍𝐂𝐂 graph property, we show the two connected components of the final graph. The constraint πšŒπš’πšŒπš•πšŽ_πšŒπšŠπš›πš_πš˜πš—_πš™πšŠπšπš‘ holds since all the vertices belong to a circuit (i.e., 𝐍𝐓𝐑𝐄𝐄 = 0) and since for each set of three consecutive vertices, colour 1 occurs at least once and at most twice (i.e.,Β the πšŠπš–πš˜πš—πš_πš•πš˜πš _πšžπš™ constraint holds).

Figure 5.90.1. Initial and final graph of the πšŒπš’πšŒπš•πšŽ_πšŒπšŠπš›πš_πš˜πš—_πš™πšŠπšπš‘ constraint
ctrs/cycle_card_on_pathActrs/cycle_card_on_pathB
(a) (b)