5.44. bipartite

DESCRIPTIONLINKSGRAPH
Origin

[Dooms06]

Constraint

πš‹πš’πš™πšŠπš›πšπš’πšπšŽ(π™½π™Ύπ™³π™΄πš‚)

Argument
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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

Consider a digraph G described by the π™½π™Ύπ™³π™΄πš‚ collection. Select a subset of arcs of G so that the corresponding graph is symmetric (i.e.,Β if there is an arc from i to j, there is also an arc from j to i) and bipartite (i.e.,Β there is no cycle involving an odd number of vertices).

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

The πš‹πš’πš™πšŠπš›πšπš’πšπšŽ constraint holds since the π™½π™Ύπ™³π™΄πš‚ collection depicts a symmetric graph with no cycle involving an odd number of vertices. The corresponding graph is depicted by FigureΒ 5.44.1.

Figure 5.44.1. The bipartite graph associated with the example
ctrs/bipartite1
Typical
|π™½π™Ύπ™³π™΄πš‚|>2
Symmetry

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

Algorithm

The sketch of a filtering algorithm for the πš‹πš’πš™πšŠπš›πšπš’πšπšŽ constraint is given inΒ [Dooms06]. Beside enforcing the fact that the graph is symmetric, it checks that the subset of mandatory vertices and arcs is bipartite and removes all potential arcs that would make the previous graph non -bipartite.

See also

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

Keywords

constraint arguments: constraint involving set variables.

constraint type: graph constraint.

filtering: DFS-bottleneck.

final graph structure: bipartite, symmetric.

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 class
β€’ πš‚πšˆπ™Όπ™Όπ™΄πšƒπšπ™Έπ™²
β€’ π™±π™Έπ™Ώπ™°πšπšƒπ™Έπšƒπ™΄

Graph model

Part (A) of Figure 5.44.2 shows the initial graph from which we 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. Part (B) of Figure 5.44.2 gives the final graph associated with the Example slot.

Figure 5.44.2. Initial and final graph of the πš‹πš’πš™πšŠπš›πšπš’πšπšŽ set constraint
ctrs/bipartiteActrs/bipartiteB
(a) (b)