## 5.239. nvalue_on_intersection

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½\pi  \pi °\pi »},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\right)$

Arguments
 $\mathrm{\pi ½\pi  \pi °\pi »}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ½\pi  \pi °\pi »}\beta ₯0$ $\mathrm{\pi ½\pi  \pi °\pi »}\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|$ $\mathrm{\pi ½\pi  \pi °\pi »}\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi \pi \pi }\right)$
Purpose

$\mathrm{\pi ½\pi  \pi °\pi »}$ is the number of distinct values that both occur in the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ collections.

Example
$\left(\begin{array}{c}2,β©1,9,1,5βͺ,\hfill \\ β©\begin{array}{c}\mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-9,\hfill \\ \mathrm{\pi \pi \pi }-9,\hfill \\ \mathrm{\pi \pi \pi }-6,\hfill \\ \mathrm{\pi \pi \pi }-9\hfill \end{array}βͺ\hfill \end{array}\right)$

Note that the two collections $\beta ©1,9,1,5\beta ͺ$ and $\beta ©2,1,9,9,6,9\beta ͺ$ share two values in common (i.e.,Β values 1 and 9). Consequently the $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi ½\pi  \pi °\pi »}$ is set to 2.

Symmetries
• Arguments are permutable w.r.t. permutation $\left(\mathrm{\pi ½\pi  \pi °\pi »}\right)$ $\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\right)$.

• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ are permutable.

• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ are permutable.

• All occurrences of two distinct values in $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ or $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ can be swapped; all occurrences of a value in $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ or $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ can be renamed to any unused value.

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$

Arc generator
$\mathrm{\pi \pi  \pi \pi ·\pi \pi Ά\pi }$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }$
Graph property(ies)
$\mathrm{\pi \pi \pi }$$=\mathrm{\pi ½\pi  \pi °\pi »}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.239.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi }$ graph property we show the connected components of the final graph. The variable $\mathrm{\pi ½\pi  \pi °\pi »}$ is equal to this number of connected components. Note that all the vertices corresponding to the variables that take values 5, 2 or 6 were removed from the final graph since there is no arc for which the associated equality constraint holds.