5.264. path

DESCRIPTIONLINKSGRAPH
Origin

Derived from πš‹πš’πš—πšŠπš›πš’_πšπš›πšŽπšŽ.

Constraint

πš™πšŠπšπš‘(π™½π™Ώπ™°πšƒπ™·,π™½π™Ύπ™³π™΄πš‚)

Arguments
π™½π™Ώπ™°πšƒπ™·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›)
Restrictions
π™½π™Ώπ™°πšƒπ™·β‰₯1
π™½π™Ώπ™°πšƒπ™·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™½π™Ύπ™³π™΄πš‚|>0
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

Cover the digraph G described by the π™½π™Ύπ™³π™΄πš‚ collection with π™½π™Ώπ™°πšƒπ™· paths in such a way that each vertex of G belongs to exactly one path.

Example
3,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-1,πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-3,πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-5,πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-7,πš’πš—πšπšŽπš‘-5𝚜𝚞𝚌𝚌-1,πš’πš—πšπšŽπš‘-6𝚜𝚞𝚌𝚌-6,πš’πš—πšπšŽπš‘-7𝚜𝚞𝚌𝚌-7,πš’πš—πšπšŽπš‘-8𝚜𝚞𝚌𝚌-6

The πš™πšŠπšπš‘ constraint holds since its second argument corresponds to the 3 (i.e.,Β the first argument of the πš™πšŠπšπš‘ constraint) paths depicted by FigureΒ 5.264.1.

Figure 5.264.1. The three paths corresponding to the Example slot
ctrs/path1
Symmetry

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

Reformulation

The πš™πšŠπšπš‘ constraint can be expressed in term of (1)Β a set of |π™½π™Ύπ™³π™΄πš‚| 2 reified constraints for avoiding circuit between more than one node and of (2)Β |π™½π™Ύπ™³π™΄πš‚| reified constraints and of one sum constraint for counting the paths and of (3)Β a set of |π™½π™Ύπ™³π™΄πš‚| 2 reified constraints and of |π™½π™Ύπ™³π™΄πš‚| inequalities constraints for enforcing the fact that each vertex has at most two children.

  1. For each vertex π™½π™Ύπ™³π™΄πš‚[i] (i∈[1,|π™½π™Ύπ™³π™΄πš‚|]) of the π™½π™Ύπ™³π™΄πš‚ collection we create a variable R i that takes its value within interval [1,|π™½π™Ύπ™³π™΄πš‚|]. This variable represents the rank of vertex π™½π™Ύπ™³π™΄πš‚[i] within a solution. It is used to prevent the creation of circuit involving more than one vertex as explained now. For each pair of vertices π™½π™Ύπ™³π™΄πš‚[i],π™½π™Ύπ™³π™΄πš‚[j] (i,j∈[1,|π™½π™Ύπ™³π™΄πš‚|]) of the π™½π™Ύπ™³π™΄πš‚ collection we create a reified constraint of the form π™½π™Ύπ™³π™΄πš‚[i].𝚜𝚞𝚌𝚌=π™½π™Ύπ™³π™΄πš‚[j].πš’πš—πšπšŽπš‘βˆ§iβ‰ jβ‡’R i <R j . The purpose of this constraint is to express the fact that, if there is an arc from vertex π™½π™Ύπ™³π™΄πš‚[i] to another vertex π™½π™Ύπ™³π™΄πš‚[j], then R i should be strictly less than R j .

  2. For each vertex π™½π™Ύπ™³π™΄πš‚[i] (i∈[1,|π™½π™Ύπ™³π™΄πš‚|]) of the π™½π™Ύπ™³π™΄πš‚ collection we create a 0-1 variable B i and state the following reified constraint π™½π™Ύπ™³π™΄πš‚[i].𝚜𝚞𝚌𝚌=π™½π™Ύπ™³π™΄πš‚[i].πš’πš—πšπšŽπš‘β‡”B i in order to force variable B i to be set to value 1 if and only if there is a loop on vertex π™½π™Ύπ™³π™΄πš‚[i]. Finally we create a constraint π™½π™Ώπ™°πšƒπ™·=B 1 +B 2 +...+B |π™½π™Ύπ™³π™΄πš‚| for stating the fact that the number of paths is equal to the number of loops of the graph.

  3. For each pair of vertices π™½π™Ύπ™³π™΄πš‚[i],π™½π™Ύπ™³π™΄πš‚[j] (i,j∈[1,|π™½π™Ύπ™³π™΄πš‚|]) of the π™½π™Ύπ™³π™΄πš‚ collection we create a 0-1 variable B ij and state the following reified constraint π™½π™Ύπ™³π™΄πš‚[i].𝚜𝚞𝚌𝚌=π™½π™Ύπ™³π™΄πš‚[j].πš’πš—πšπšŽπš‘βˆ§iβ‰ j⇔B ij . Variable B ij is set to value 1 if and only if there is an arc from π™½π™Ύπ™³π™΄πš‚[i] to π™½π™Ύπ™³π™΄πš‚[j]. Then for each vertex π™½π™Ύπ™³π™΄πš‚[j] (j∈[1,|π™½π™Ύπ™³π™΄πš‚|]) we create a constraint of the form B 1j +B 2j +...+B |π™½π™Ύπ™³π™΄πš‚|j ≀2.

See also

common keyword: πšŒπš’πš›πšŒπšžπš’πšΒ (graph partitioning constraint,one_succ), 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šŠπšπš‘_πšπš›πš˜πš–_𝚝𝚘 (path, select an induced subgraph so that there is a path from a given vertex to an other given vertex).

generalisation: πš‹πš’πš—πšŠπš›πš’_πšπš›πšŽπšŽΒ (at most one child replaced by at most two children), πšπšŽπš–πš™πš˜πš›πšŠπš•_πš™πšŠπšπš‘Β (vertices are located in time, and to each arc corresponds a precedence constraint), πšπš›πšŽπšŽΒ (at most one child replaced by no limit on the number of children).

implies: πš‹πš’πš—πšŠπš›πš’_πšπš›πšŽπšŽ.

Keywords

combinatorial object: path.

constraint type: graph constraint, graph partitioning constraint.

final graph structure: connected component, tree, 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€’ πŒπ€π—_𝐍𝐒𝐂𝐂≀1
β€’ 𝐍𝐂𝐂=π™½π™Ώπ™°πšƒπ™·
β€’ πŒπ€π—_πˆπƒ=1

Graph class
𝙾𝙽𝙴_πš‚πš„π™²π™²

Graph model

We use the same graph constraint as for the πš‹πš’πš—πšŠπš›πš’_πšπš›πšŽπšŽ constraint, except that we replace the graph property πŒπ€π—_πˆπƒβ‰€2, which constraints the maximum in-degree of the final graph to not exceed 2 by πŒπ€π—_πˆπƒ=1. πŒπ€π—_πˆπƒ does not consider loops: This is why we do not have any problem with the final node of each path.

PartsΒ (A) andΒ (B) of FigureΒ 5.264.2 respectively show the initial and final graph associated with the Example slot. Since we use the 𝐍𝐂𝐂 graph property, we display the three connected components of the final graph. Each of them corresponds to a path. Since we use the πŒπ€π—_πˆπƒ graph property, we also show with a double circle a vertex that has a maximum number of predecessors.

The πš™πšŠπšπš‘ constraint holds since all strongly connected components of the final graph have no more than one vertex, since π™½π™Ώπ™°πšƒπ™·=𝐍𝐂𝐂=3 and since πŒπ€π—_πˆπƒ=1.

Figure 5.264.2. Initial and final graph of the πš™πšŠπšπš‘ constraint
ctrs/pathActrs/pathB
(a) (b)