## 5.378. strict_lex2

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi ‘}\mathtt{2}\left(\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }\right)$

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 of domain variables, enforces that both adjacent rows, and adjacent columns are lexicographically ordered (adjacent rows and adjacent columns cannot be equal).

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

The $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi ‘}\mathtt{2}$ constraint holds since:

• The first row $\beta ©2,2,3\beta ͺ$ is lexicographically strictly less than the second row $\beta ©2,3,1\beta ͺ$.

• The first column $\beta ©2,2\beta ͺ$ is lexicographically strictly less than the second column $\beta ©2,3\beta ͺ$.

• The second column $\beta ©2,3\beta ͺ$ is lexicographically strictly less than the third column $\beta ©3,1\beta ͺ$.

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.

Reformulation

The $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi ‘}\mathtt{2}$ constraint can be expressed as a conjunction of two $\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ constraints: A first $\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ constraint on the $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }$ argument and a second $\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ constraint on the transpose of the $\mathrm{\pi Ό\pi °\pi \pi \pi Έ\pi }$ argument.

Systems