## 5.63. common_interval

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\left(\begin{array}{c}\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{1},\hfill \\ \mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{2},\hfill \\ \mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\hfill \\ \mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\hfill \\ \mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\hfill \end{array}\right)$

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

$\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{1}$ is the number of variables of the collection of variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ taking a value in one of the intervals derived from the values assigned to the variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$: To each value $v$ assigned to a variable of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ we associate the interval $\left[\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·\beta v/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\beta ,\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·\beta v/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\beta +\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}-1\right]$.

$\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{2}$ is the number of variables of the collection of variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ taking a value in one of the intervals derived from the values assigned to the variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$: To each value $v$ assigned to a variable of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ we associate the interval $\left[\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·\beta v/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\beta ,\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·\beta v/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\beta +\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}-1\right]$.

Example
$\left(\begin{array}{c}3,2,β©8,6,6,0βͺ,\hfill \\ β©\begin{array}{c}\mathrm{\pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-7\hfill \end{array}βͺ,3\hfill \end{array}\right)$

In the example, the last argument $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}=3$ defines the following family of intervals $\left[3Β·k,3Β·k+2\right]$, where $k$ is an integer. As a consequence the items of collection $\beta ©8,6,6,0\beta ͺ$ respectively correspond to intervals $\left[6,8\right]$, $\left[6,8\right]$, $\left[6,8\right]$ and $\left[0,2\right]$. Similarly the items of collection $\beta ©7,3,3,3,3,7\beta ͺ$ respectively correspond to intervals $\left[6,8\right]$, $\left[3,5\right]$, $\left[3,5\right]$, $\left[3,5\right]$, $\left[3,5\right]$, $\left[6,8\right]$. The $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since:

• Its first argument $\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{1}=3$ is the number of intervals associated with the items of collection $\beta ©8,6,6,0\beta ͺ$ that also correspond to intervals associated with $\beta ©7,3,3,3,3,7\beta ͺ$.

• Its second argument $\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{2}=2$ is the number of intervals associated with the items of collection $\beta ©7,3,3,3,3,7\beta ͺ$ that also correspond to intervals associated with $\beta ©8,6,6,0\beta ͺ$.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }\right)>1$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}>1$
Symmetries
• Arguments are permutable w.r.t. permutation $\left(\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{1},\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{2}\right)$ $\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\right)$ $\left(\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\right)$.

• 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.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ that belongs to the $k$-th interval, of size $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$, can be replaced by any other value of the same interval.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ that belongs to the $k$-th interval, of size $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$, can be replaced by any other value of the same interval.

specialisation: $\mathrm{\pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }/\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$).

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)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$
Graph property(ies)
 $\beta ’$$\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$=\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{1}$ $\beta ’$$\mathrm{\pi \pi \pi \pi \pi }$$=\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{2}$

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

PartsΒ (A) andΒ (B) of FigureΒ 5.63.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ and $\mathrm{\pi \pi \pi \pi \pi }$ graph properties, the source and sink vertices of the final graph are stressed with a double circle. Since the graph has only 3 sources and 2 sinks the variables $\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{1}$ and $\mathrm{\pi ½\pi ²\pi Ύ\pi Ό\pi Ό\pi Ύ\pi ½}\mathtt{2}$ are respectively equal to 3 and 2. Note that the vertices corresponding to the variables that take values 0 or 3 were removed from the final graph since there is no arc for which the associated arc constraint holds.