## 5.242. nvector

Origin

Introduced by G.Β Chabert as a generalisation of $\mathrm{\pi \pi \pi \pi \pi \pi }$

Constraint

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

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
 $\mathrm{\pi ½\pi  \pi ΄\pi ²}\beta ₯\mathrm{\pi \pi \pi }\left(1,|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|\right)$ $\mathrm{\pi ½\pi  \pi ΄\pi ²}\beta €|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|$ $\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

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

Example
$\left(\begin{array}{c}2,β©\begin{array}{c}\mathrm{\pi \pi \pi }-β©5,6βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©5,6βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©9,3βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©5,6βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©9,3βͺ\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi ½\pi  \pi ΄\pi ²}=2$ is set to the number of distinct tuples of values (i.e.,Β tuples $\beta ©5,6\beta ͺ$ and $\beta ©9,3\beta ͺ$) occurring within the collection $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$. FigureΒ 5.242.1 depicts with a thick rectangle a possible initial domain for each of the five vectors and with a grey circle each tuple of values of the corresponding solution.

Symmetries
• 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.

Remark

It was shown inΒ [ChabertLorca09], [ChabertJaulinLorca09] that, finding out whether a $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint has a solution or not is NP-hard (i.e.,Β the restriction to the rectangle case and to the atmost side of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ were considered for this purpose). This was achieved by reduction from the rectangle clique partition problem.

Reformulation

Assume the collection $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ is not empty (otherwise $\mathrm{\pi ½\pi  \pi ΄\pi ²}=0$). In this context, let $n$ and $m$ respectively denote the number of vectors of the collection $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ and the number of components of each vector. Furthermore, let ${\mathrm{\Xi ±}}_{i}=min\left(\underset{Μ²}{{C}_{1i}},\underset{Μ²}{{C}_{2i}},...,\underset{Μ²}{{C}_{ni}}\right)$, ${\mathrm{\Xi ²}}_{i}=max\left(\stackrel{Β―}{{C}_{1i}},\stackrel{Β―}{{C}_{2i}},...,\stackrel{Β―}{{C}_{ni}}\right)$, ${\mathrm{\Xi ³}}_{i}={\mathrm{\Xi ²}}_{i}-{\mathrm{\Xi ±}}_{i}+1$, $\left(i\beta \left[1,m\right]\right)$. By associating to each vector

$\beta ©{C}_{k1},{C}_{k2},...,{C}_{km}\beta ͺ,\left(k\beta \left[1,n\right]\right)$

a variable

${D}_{k}=\underset{1\beta €i\beta €m}{\beta }\left(\left(\underset{i

the constraint

Β Β Β $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi  \pi ΄\pi ²},$

Β Β Β Β Β Β Β Β Β Β Β Β Β $\beta ©\mathrm{\pi \pi \pi }-\beta ©{C}_{11},{C}_{12},...,{C}_{1m}\beta ͺ,$

Β Β Β Β Β Β Β Β Β Β Β Β Β Β $\mathrm{\pi \pi \pi }-\beta ©{C}_{21},{C}_{22},...,{C}_{2m}\beta ͺ,$

Β Β Β Β Β Β Β Β Β Β Β Β Β Β $........................$

Β Β Β Β Β Β Β Β Β Β Β Β Β Β $\mathrm{\pi \pi \pi }-\beta ©{C}_{n1},{C}_{n2},...,{C}_{nm}\beta ͺ\beta ͺ\right)$

can be expressed in term of the constraint

Β Β Β $\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi  \pi ΄\pi ²},\beta ©{D}_{1},{D}_{2},...,{D}_{n}\beta ͺ\right)$.

Note that the previous reformulation does not work anymore if the variables have a continuous domain, or if an overflow occurs while propagating the equality constraint ${D}_{k}={\beta }_{1\beta €i\beta €m}\left(\left({\beta }_{i (i.e.,Β the number of components $m$ is too big).

When using this reformulation with respect to the Example slot we first introduce ${D}_{1}=1Β·6-3+\left(4Β·5-20\right)\right)=3$, ${D}_{2}=1Β·6-3+\left(4Β·5-20\right)\right)=3$, ${D}_{3}=1Β·3-3+\left(4Β·9-20\right)\right)=16$, ${D}_{4}=1Β·6-3+\left(4Β·5-20\right)\right)=3$, ${D}_{5}=1Β·3-3+\left(4Β·9-20\right)\right)=16$ and then get the constraint $\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(2,\beta ©3,3,16,3,16\beta ͺ\right)$.

generalisation: $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$Β (replace an equality with the number of distinct vectors by a comparison with the number of distinct nvectors).

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

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 }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\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 5.242.2 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.