## 5.251. open_minimum

Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi Ό\pi Έ\pi ½},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

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

$\mathrm{\pi Ό\pi Έ\pi ½}$ is the minimum value of the variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }$, $\left(1\beta €i\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right)$ for which $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi \pi }=1$ (at least one of the Boolean variables is set to 1).

Example
$\left(\begin{array}{c}3,β©\begin{array}{cc}\mathrm{\pi \pi \pi }-3\hfill & \mathrm{\pi \pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-1\hfill & \mathrm{\pi \pi \pi \pi }-0,\hfill \\ \mathrm{\pi \pi \pi }-7\hfill & \mathrm{\pi \pi \pi \pi }-0,\hfill \\ \mathrm{\pi \pi \pi }-5\hfill & \mathrm{\pi \pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-5\hfill & \mathrm{\pi \pi \pi \pi }-1\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi Ό\pi Έ\pi ½}=3$ is set to the minimum value of values $3,1,7,5,5$ for which the corresponding Boolean $1,0,0,1,1$ is set to 1 (i.e.,Β values $3,5,5$).

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

• One and the same constant can be added to $\mathrm{\pi Ό\pi Έ\pi ½}$ as well as to the $\mathrm{\pi \pi \pi }$ attribute of all items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Remark

The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint is used in the reformulation of the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint.

FigureΒ 5.251.1 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint. Let ${\mathrm{\pi  \pi °\pi }}_{i},{\mathrm{\pi ±}}_{i}$ be the ${i}^{th}$ item of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection. To each triple $\left(\mathrm{\pi Ό\pi Έ\pi ½},{\mathrm{\pi  \pi °\pi }}_{i},{\mathrm{\pi ±}}_{i}\right)$ corresponds a signature variable ${\mathrm{\pi }}_{i}$ as well as the following signature constraint: $\left({\mathrm{\pi ±}}_{i}=1\beta §\mathrm{\pi Ό\pi Έ\pi ½}<{\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=0\right)\beta §\left({\mathrm{\pi ±}}_{i}=1\beta §\mathrm{\pi Ό\pi Έ\pi ½}={\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=1\right)\beta §\left({\mathrm{\pi ±}}_{i}=1\beta §\mathrm{\pi Ό\pi Έ\pi ½}>{\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=2\right)\beta §\left({\mathrm{\pi ±}}_{i}=0\beta §\mathrm{\pi Ό\pi Έ\pi ½}<{\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=3\right)\beta §\left({\mathrm{\pi ±}}_{i}=0\beta §\mathrm{\pi Ό\pi Έ\pi ½}={\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=4\right)\beta §\left({\mathrm{\pi ±}}_{i}=0\beta §\mathrm{\pi Ό\pi Έ\pi ½}>{\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=5\right)$.