5.323. strongly_connected

DESCRIPTIONLINKSGRAPH
Origin

[AlthausBockmayrElfKasperJungerMehlhorn02]

Constraint

πšœπšπš›πš˜πš—πšπš•πš’_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš(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‘)
Purpose

Consider a digraph G described by the π™½π™Ύπ™³π™΄πš‚ collection. Select a subset of arcs of G so that we have one single strongly connected component involving all vertices of G.

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

The πšœπšπš›πš˜πš—πšπš•πš’_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš constraint holds since the π™½π™Ύπ™³π™΄πš‚ collection depicts a graph involving one single strongly connected component (i.e.,Β since we have a circuit visiting successively the vertices 1, 2, 3, 5, and 4).

Symmetry

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

Algorithm

The sketch of a filtering algorithm for the πšœπšπš›πš˜πš—πšπš•πš’_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš constraint is given inΒ [Dooms06].

See also

common keyword: 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).

implied by: πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš.

related: πšŒπš’πš›πšŒπšžπš’πšΒ (one single strongly connected component in the final solution).

Keywords

constraint arguments: constraint involving set variables.

constraint type: graph constraint.

filtering: linear programming.

final graph structure: strongly connected component.

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)
𝐌𝐈𝐍_𝐍𝐒𝐂𝐂=|π™½π™Ύπ™³π™΄πš‚|

Graph model

PartΒ (A) of FigureΒ 5.323.1 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. PartΒ (B) of FigureΒ 5.323.1 gives the final graph associated with the Example slot. The πšœπšπš›πš˜πš—πšπš•πš’_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš constraint holds since the final graph contains one single strongly connected component mentioning every vertex of the initial graph.

Figure 5.323.1. Initial and final graph of the πšœπšπš›πš˜πš—πšπš•πš’_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš set constraint
ctrs/strongly_connectedActrs/strongly_connectedB
(a) (b)
Signature

Since the maximum number of vertices of the final graph is equal to |π™½π™Ύπ™³π™΄πš‚| we can rewrite the graph property 𝐌𝐈𝐍_𝐍𝐒𝐂𝐂=|π™½π™Ύπ™³π™΄πš‚| to 𝐌𝐈𝐍_𝐍𝐒𝐂𝐂β‰₯|π™½π™Ύπ™³π™΄πš‚| and simplify 𝐌𝐈𝐍_𝐍𝐒𝐂𝐂 Β― Μ² to 𝐌𝐈𝐍_𝐍𝐒𝐂𝐂 Β―.