5.15. allperm

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }\right)$

Synonyms

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$.

Type
 $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Argument
 $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi },\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi },\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi £\pi }$$\left(\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

Given a matrix of domain variables, enforces that the first row is lexicographically less than or equal to all permutations of all other rows.

Example
$\left(\begin{array}{c}β©\begin{array}{c}\mathrm{\pi \pi \pi }-β©1,2,3βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©3,1,2βͺ\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since vector $\beta ©1,2,3\beta ͺ$ is lexicographically less than or equal to all the permutations of vector $\beta ©3,1,2\beta ͺ$ (i.e.,Β $\beta ©1,2,3\beta ͺ$, $\beta ©1,3,2\beta ͺ$, $\beta ©2,1,3\beta ͺ$, $\beta ©2,3,1\beta ͺ$, $\beta ©3,1,2\beta ͺ$, $\beta ©3,2,1\beta ͺ$).

Typical
 $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|>1$ $|\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }|>1$
Symmetry

One and the same constant can be added to the $\mathrm{\pi \pi \pi }$ attribute of all items of $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }.\mathrm{\pi \pi \pi }$.

Usage

A symmetry-breaking constraint.

Keywords
Arc input(s)

$\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\left(<\right)\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{2}\right)$

Arc arity
Arc constraint(s)
 $\beta ’\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{1}.\mathrm{\pi \pi \pi ’}=1$ $\beta ’\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{2}.\mathrm{\pi \pi \pi ’}>1$ $\beta ’$$\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }|-1$

Graph class
 $\beta ’$$\mathrm{\pi °\pi ²\pi \pi ²\pi »\pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$ $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$

Graph model

We generate a graph with an arc constraint $\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ between the vertex corresponding to the first item of the $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }$ collection and the vertices associated with all other items of the $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }$ collection. This is achieved by specifying that (1)Β an arc should start from the first item (i.e.,Β $\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{1}.\mathrm{\pi \pi \pi ’}=1$) and (2)Β an arc should not end on the first item (i.e.,Β $\mathrm{\pi \pi \pi \pi \pi \pi ‘}\mathtt{2}.\mathrm{\pi \pi \pi ’}>1$). We finally state that all these arcs should belong to the final graph. PartsΒ (A) andΒ (B) of FigureΒ 5.15.1 respectively show the initial and final graph associated with the Example slot.