## 5.214. min_n

Origin
Constraint

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

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

$\mathrm{\pi Ό\pi Έ\pi ½}$ is the minimum value of rank $\mathrm{\pi \pi °\pi ½\pi Ί}$ (i.e.,Β the ${\mathrm{\pi \pi °\pi ½\pi Ί}}^{th}$ smallest distinct value) of the collection of domain variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$. Sources have a rank of 0.

Example
$\left(3,1,β©3,1,7,1,6βͺ\right)$

The $\mathrm{\pi \pi \pi }_\mathrm{\pi }$ constraint holds since its first argument $\mathrm{\pi Ό\pi Έ\pi ½}=3$ is fixed to the second (i.e.,Β $\mathrm{\pi \pi °\pi ½\pi Ί}+1$) smallest distinct value of the collection $\beta ©3,1,7,1,6\beta ͺ$. Note that identical values are only counted once: this is why the minimum of order 1 is 3 instead of 1.

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

Algorithm
Reformulation

The constraint $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$$\left(1,\beta ©\mathrm{\pi Ό\pi Έ\pi ½}\beta ͺ,\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$ enforces $\mathrm{\pi Ό\pi Έ\pi ½}$ to be assigned one of the values of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$. The constraint $\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi  \pi °\pi »},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$ provides a hand on the number of distinct values assigned to the variables of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$. By associating to each variable ${V}_{i}$ $\left(i\beta \left[1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right]\right)$ of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection a rank variable ${R}_{i}\beta \left[0,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-1\right]$ with the reified constraint ${R}_{i}=\mathrm{\pi \pi °\pi ½\pi Ί}\beta {V}_{i}=\mathrm{\pi Ό\pi Έ\pi ½}$, the inequality ${R}_{i}<\mathrm{\pi ½\pi  \pi °\pi »}$, and by creating for each pair of variables ${V}_{i},{V}_{j}$ $\left(i,j the reified constraints

Β Β Β ${V}_{i}<{V}_{j}\beta {R}_{i}<{R}_{j}$,

Β Β Β ${V}_{i}={V}_{j}\beta {R}_{i}={R}_{j}$,

Β Β Β ${V}_{i}>{V}_{j}\beta {R}_{i}>{R}_{j}$,

one can reformulate the $\mathrm{\pi \pi \pi }_\mathrm{\pi }$ constraint in term of $3Β·\frac{|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|Β·\left(|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-1\right)}{2}+1$ reified constraints.

generalisation: $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$Β (absolute minimum replaced by minimum or order $\mathrm{\pi }$).

Keywords
Arc input(s)

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

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 \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\beta \left(\begin{array}{c}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi ’}=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi ’},\hfill \\ \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }<\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\hfill \end{array}\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi °\pi ½\pi Ί},\mathrm{\pi Ό\pi °\pi \pi Έ\pi ½\pi },\mathrm{\pi \pi \pi }\right)=\mathrm{\pi Ό\pi Έ\pi ½}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.214.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi }$ graph property, the vertex of rank 1 (without considering the loops) of the final graph is shown in grey.

Automaton

FigureΒ 5.214.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi }_\mathrm{\pi }$ constraint. FigureΒ 5.214.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi }_\mathrm{\pi }$ constraint. To each item of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a signature variable ${\mathrm{\pi }}_{i}$ that is equal to 1.