## 5.246. open_atleast

Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi },\mathrm{\pi ½},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$

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

Let $\mathrm{\pi ±}$ be the variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ for which the corresponding position belongs to the set $\mathrm{\pi }$. Positions are numbered from 1. At least $\mathrm{\pi ½}$ variables of $\mathrm{\pi ±}$ are assigned value $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.

Example
$\left(\begin{array}{c}\left\{2,3,4\right\},2,\hfill \\ β©4,2,4,4βͺ,4\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since, within the last three (i.e.,Β $\mathrm{\pi }=\left\{2,3,4\right\}$) values of the collection $\beta ©4,2,4,4\beta ͺ$, at least $\mathrm{\pi ½}=2$ values are equal to value $\mathrm{\pi  \pi °\pi »\pi \pi ΄}=4$.

Symmetries
• $\mathrm{\pi ½}$ can be decreased to any value $\beta ₯0$.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ that is different from $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ can be replaced by any other value.

Keywords
Arc input(s)

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

Arc generator
$\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
 $\beta ’\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }=\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ $\beta ’$$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi ’},\mathrm{\pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$\beta ₯\mathrm{\pi ½}$

Graph model

Since each arc constraint involves only one vertex ($\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ is fixed), we employ the $\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$ arc generator in order to produce a graph with a single loop on each vertex. Variables for which the corresponding position does not belong to the set $\mathrm{\pi }$ are removed from the final graph by the second condition of the arc -constraint.

PartsΒ (A) andΒ (B) of FigureΒ 5.246.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the loops of the final graph are stressed in bold.