5.58. clique

DESCRIPTIONLINKSGRAPH
Origin

[Fahle02]

Constraint

๐šŒ๐š•๐š’๐šš๐šž๐šŽ(๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด,๐™ฝ๐™พ๐™ณ๐™ด๐š‚)

Arguments
๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด๐š๐šŸ๐šŠ๐š›
๐™ฝ๐™พ๐™ณ๐™ด๐š‚๐šŒ๐š˜๐š•๐š•๐šŽ๐šŒ๐š๐š’๐š˜๐š—(๐š’๐š—๐š๐šŽ๐šก-๐š’๐š—๐š,๐šœ๐šž๐šŒ๐šŒ-๐šœ๐šŸ๐šŠ๐š›)
Restrictions
๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ดโ‰ฅ0
๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ดโ‰ค|๐™ฝ๐™พ๐™ณ๐™ด๐š‚|
๐š›๐šŽ๐šš๐šž๐š’๐š›๐šŽ๐š(๐™ฝ๐™พ๐™ณ๐™ด๐š‚,[๐š’๐š—๐š๐šŽ๐šก,๐šœ๐šž๐šŒ๐šŒ])
๐™ฝ๐™พ๐™ณ๐™ด๐š‚.๐š’๐š—๐š๐šŽ๐šกโ‰ฅ1
๐™ฝ๐™พ๐™ณ๐™ด๐š‚.๐š’๐š—๐š๐šŽ๐šกโ‰ค|๐™ฝ๐™พ๐™ณ๐™ด๐š‚|
๐š๐š’๐šœ๐š๐š’๐š—๐šŒ๐š(๐™ฝ๐™พ๐™ณ๐™ด๐š‚,๐š’๐š—๐š๐šŽ๐šก)
๐™ฝ๐™พ๐™ณ๐™ด๐š‚.๐šœ๐šž๐šŒ๐šŒโ‰ฅ1
๐™ฝ๐™พ๐™ณ๐™ด๐š‚.๐šœ๐šž๐šŒ๐šŒโ‰ค|๐™ฝ๐™พ๐™ณ๐™ด๐š‚|
Purpose

Consider a digraph G described by the ๐™ฝ๐™พ๐™ณ๐™ด๐š‚ collection: to the i th item of the ๐™ฝ๐™พ๐™ณ๐™ด๐š‚ collection corresponds the i th vertex of G; To each value j of the i th ๐šœ๐šž๐šŒ๐šŒ variable corresponds an arc from the i th vertex to the j th vertex. Select a subset ๐’ฎ of the vertices of G that forms a clique of size ๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด (i.e.,ย there is an arc between each pair of distinct vertices of ๐’ฎ).

Example
3,๐š’๐š—๐š๐šŽ๐šก-1๐šœ๐šž๐šŒ๐šŒ-โˆ…,๐š’๐š—๐š๐šŽ๐šก-2๐šœ๐šž๐šŒ๐šŒ-{3,5},๐š’๐š—๐š๐šŽ๐šก-3๐šœ๐šž๐šŒ๐šŒ-{2,5},๐š’๐š—๐š๐šŽ๐šก-4๐šœ๐šž๐šŒ๐šŒ-โˆ…,๐š’๐š—๐š๐šŽ๐šก-5๐šœ๐šž๐šŒ๐šŒ-{2,3}

The ๐šŒ๐š•๐š’๐šš๐šž๐šŽ constraint holds since the ๐™ฝ๐™พ๐™ณ๐™ด๐š‚ collection depicts a clique involving 3 vertices (namely vertices 2, 3 and 5) and since its first argument ๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด is set to the number of vertices of this clique.

Typical
๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ดโ‰ฅ2
๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด<|๐™ฝ๐™พ๐™ณ๐™ด๐š‚|
|๐™ฝ๐™พ๐™ณ๐™ด๐š‚|>2
Symmetry

Items of ๐™ฝ๐™พ๐™ณ๐™ด๐š‚ are permutable.

Algorithm

[Fahle02],ย [Regin03CPAIOR], [Regin03]. The algorithm for finding maximum cliques in an undirected graph of C.ย Bron and J.ย Kerboschย [BronKerbosch73] was adapted by J.-C.ย Rรฉgin to the context of constraint programming in his papers.

See also

common keyword: ๐š•๐š’๐š—๐š”_๐šœ๐šŽ๐š_๐š๐š˜_๐š‹๐š˜๐š˜๐š•๐šŽ๐šŠ๐š—๐šœย (constraint involving set variables, can be used for channelling).

used in graph description: ๐š’๐š—_๐šœ๐šŽ๐š.

Keywords

constraint arguments: constraint involving set variables.

constraint type: graph constraint.

final graph structure: symmetric.

problems: maximum clique.

Arc input(s)

๐™ฝ๐™พ๐™ณ๐™ด๐š‚

Arc generator
๐ถ๐ฟ๐ผ๐‘„๐‘ˆ๐ธ(โ‰ )โ†ฆ๐šŒ๐š˜๐š•๐š•๐šŽ๐šŒ๐š๐š’๐š˜๐š—(๐š—๐š˜๐š๐šŽ๐šœ1,๐š—๐š˜๐š๐šŽ๐šœ2)

Arc arity
Arc constraint(s)
๐š’๐š—_๐šœ๐šŽ๐š(๐š—๐š˜๐š๐šŽ๐šœ2.๐š’๐š—๐š๐šŽ๐šก,๐š—๐š˜๐š๐šŽ๐šœ1.๐šœ๐šž๐šŒ๐šŒ)
Graph property(ies)
โ€ข ๐๐€๐‘๐‚=๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด*๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด-๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด
โ€ข ๐๐•๐„๐‘๐“๐„๐—=๐š‚๐™ธ๐š‰๐™ด_๐™ฒ๐™ป๐™ธ๐š€๐š„๐™ด

Graph class
๐š‚๐šˆ๐™ผ๐™ผ๐™ด๐šƒ๐š๐™ธ๐™ฒ

Graph model

Note the use of set variables for modelling the fact that the vertices of the final graph have more than one successor: The successor variable associated with each vertex contains the successors of the corresponding vertex.

Partย (A) of Figureย 5.58.1 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 ๐šœ๐šž๐šŒ๐šŒ attribute of a given vertex. Partย (B) of Figureย 5.58.1 gives the final graph associated with the Example slot. Since we both use the ๐๐€๐‘๐‚ and ๐๐•๐„๐‘๐“๐„๐— graph properties, the arcs and the vertices of the final graph are stressed in bold. The final graph corresponds to a clique containing three vertices.

Figure 5.58.1. Initial and final graph of the ๐šŒ๐š•๐š’๐šš๐šž๐šŽ set constraint
ctrs/cliqueActrs/cliqueB
(a) (b)