## 5.42. atmost_nvector

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½\pi  \pi ΄\pi ²},\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$

Type
 $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Arguments
 $\mathrm{\pi ½\pi  \pi ΄\pi ²}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\right)$
Restrictions
 $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|\beta ₯1$ $\mathrm{\pi ½\pi  \pi ΄\pi ²}\beta ₯\mathrm{\pi \pi \pi }\left(1,|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi },\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi £\pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi },\mathrm{\pi \pi \pi }\right)$
Purpose

The number of distinct tuples of values taken by the vectors of the collection $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ is less than or equal to $\mathrm{\pi ½\pi  \pi ΄\pi ²}$. Two tuples of values $\beta ©{A}_{1},{A}_{2},\beta ―,{A}_{m}\beta ͺ$ and $\beta ©{B}_{1},{B}_{2},\beta ―,{B}_{m}\beta ͺ$ are $distinct$ if and only if there exist an integer $i\beta \left[1,m\right]$ such that .

Example
$\left(\begin{array}{c}3,β©\begin{array}{c}\mathrm{\pi \pi \pi }-β©5,6βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©5,6βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©9,3βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©5,6βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©9,3βͺ\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since the collection $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ involves at most 3 distinct tuples of values (i.e.,Β in fact the 2 distinct tuples $\beta ©5,6\beta ͺ$ and $\beta ©9,3\beta ͺ$).

Typical
 $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|>1$ $\mathrm{\pi ½\pi  \pi ΄\pi ²}>1$ $\mathrm{\pi ½\pi  \pi ΄\pi ²}<|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|>1$
Symmetries
• $\mathrm{\pi ½\pi  \pi ΄\pi ²}$ can be increased.

• Items of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ are permutable.

• Items of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }.\mathrm{\pi \pi \pi }$ are permutable (same permutation used).

• All occurrences of two distinct tuples of values of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }.\mathrm{\pi \pi \pi }$ can be swapped; all occurrences of a tuple of values of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }.\mathrm{\pi \pi \pi }$ can be renamed to any unused tuple of values.

Arg. properties

Contractible wrt. $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$.

Reformulation

By introducing an extra variable $\mathrm{\pi ½\pi  }\beta \left[0,|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|\right]$, the $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi  },\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$ constraint can be expressed in term of an $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi  },\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$ constraint and of an inequality constraint $\mathrm{\pi ½\pi  }\beta €\mathrm{\pi ½\pi  \pi ΄\pi ²}$.

implied by: $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$Β ($\beta €$ $\mathrm{\pi ½\pi  \pi ΄\pi ²}$ replaced by $=$ $\mathrm{\pi ½\pi  \pi ΄\pi ²}$), $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$.

Keywords
Arc input(s)

$\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$

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

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$\beta €\mathrm{\pi ½\pi  \pi ΄\pi ²}$

Graph class
$\mathrm{\pi ΄\pi \pi \pi Έ\pi  \pi °\pi »\pi ΄\pi ½\pi ²\pi ΄}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.42.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a tuple of values that is assigned to some vectors of the $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ collection. The 2 following tuple of values $\beta ©5,6\beta ͺ$ and $\beta ©9,3\beta ͺ$ are used by the vectors of the $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ collection.