## 5.22. 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
 $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|\beta ₯1$ $\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 $\mathrm{\beta ³}$ of domain variables, enforces that the first row is lexicographically less than or equal to all permutations of all other rows. Note that the components of a given vector of the matrix $\mathrm{\beta ³}$ may be equal.

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

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 $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }.\mathrm{\pi \pi \pi }$.

Arg. properties

Suffix-contractible wrt. $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }.\mathrm{\pi \pi \pi }$ (remove items from same position).

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 5.22.1 respectively show the initial and final graph associated with the Example slot.