5.265. path_from_to

DESCRIPTIONLINKSGRAPH
Origin

[AlthausBockmayrElfKasperJungerMehlhorn02]

Constraint

πš™πšŠπšπš‘_πšπš›πš˜πš–_𝚝𝚘(π™΅πšπ™Ύπ™Ό,πšƒπ™Ύ,π™½π™Ύπ™³π™΄πš‚)

Usual name

πš™πšŠπšπš‘

Arguments
π™΅πšπ™Ύπ™Όπš’πš—πš
πšƒπ™Ύπš’πš—π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šœπšŸπšŠπš›)
Restrictions
π™΅πšπ™Ύπ™Όβ‰₯1
π™΅πšπ™Ύπ™Όβ‰€|π™½π™Ύπ™³π™΄πš‚|
πšƒπ™Ύβ‰₯1
πšƒπ™Ύβ‰€|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™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‚|
Purpose

Select some arcs of a digraph G so that there is still a path between two given vertices of G.

Example
4,3,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-βˆ…,πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-βˆ…,πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-{5},πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-{5},πš’πš—πšπšŽπš‘-5𝚜𝚞𝚌𝚌-{2,3}

The πš™πšŠπšπš‘_πšπš›πš˜πš–_𝚝𝚘 constraint holds since within the digraph G corresponding to the item of the π™½π™Ύπ™³π™΄πš‚ collection there is a path from vertex π™΅πšπ™Ύπ™Ό=4 to vertex πšƒπ™Ύ=3: this path starts from vertex 4, enters vertex 5, and ends up in vertex 3.

Symmetry

Items of π™½π™Ύπ™³π™΄πš‚ are permutable.

See also

common keyword: πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’Β (path),

πš•πš’πš—πš”_𝚜𝚎𝚝_𝚝𝚘_πš‹πš˜πš˜πš•πšŽπšŠπš—πšœΒ (constraint involving set variables),

πš™πšŠπšπš‘, πšπšŽπš–πš™πš˜πš›πšŠπš•_πš™πšŠπšπš‘Β (path).

used in graph description: πš’πš—_𝚜𝚎𝚝.

Keywords

combinatorial object: path.

constraint arguments: constraint involving set variables.

constraint type: graph constraint.

filtering: linear programming.

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—_𝚜𝚎𝚝(πš—πš˜πšπšŽπšœ2.πš’πš—πšπšŽπš‘,πš—πš˜πšπšŽπšœ1.𝚜𝚞𝚌𝚌)
Graph property(ies)
𝐏𝐀𝐓𝐇_π…π‘πŽπŒ_π“πŽ(πš’πš—πšπšŽπš‘,π™΅πšπ™Ύπ™Ό,πšƒπ™Ύ)=1

Graph model

Within the context of the Example slot, partΒ (A) of FigureΒ 5.265.1 shows the initial graph from which we choose to start. It is derived from the set associated with each vertex. Each set describes the potential values of the 𝚜𝚞𝚌𝚌 attribute of a given vertex. PartΒ (B) of FigureΒ 5.265.1 gives the final graph associated with the Example slot. Since we use the 𝐏𝐀𝐓𝐇_π…π‘πŽπŒ_π“πŽ graph property we show on the final graph the following information:

  • The vertices that respectively correspond to the start and the end of the required path are stressed in bold.

  • The arcs on the required path are also stressed in bold.

The πš™πšŠπšπš‘_πšπš›πš˜πš–_𝚝𝚘 constraint holds since there is a path from vertex 4 to vertex 3 (4 and 3 refer to the πš’πš—πšπšŽπš‘ attribute of a vertex).

Figure 5.265.1. Initial and final graph of the πš™πšŠπšπš‘_πšπš›πš˜πš–_𝚝𝚘 set constraint
ctrs/path_from_toActrs/path_from_toB
(a) (b)
Signature

Since the maximum value returned by the graph property 𝐏𝐀𝐓𝐇_π…π‘πŽπŒ_π“πŽ is equal to 1 we can rewrite 𝐏𝐀𝐓𝐇_π…π‘πŽπŒ_π“πŽ(πš’πš—πšπšŽπš‘,π™΅πšπ™Ύπ™Ό,πšƒπ™Ύ)=1 to 𝐏𝐀𝐓𝐇_π…π‘πŽπŒ_π“πŽ(πš’πš—πšπšŽπš‘,π™΅πšπ™Ύπ™Ό,πšƒπ™Ύ)β‰₯1. Therefore we simplify 𝐏𝐀𝐓𝐇_π…π‘πŽπŒ_π“πŽ Β― Μ² to 𝐏𝐀𝐓𝐇_π…π‘πŽπŒ_π“πŽ Β―.