## 5.141. global_cardinality_no_loop

 DESCRIPTION LINKS GRAPH
Origin
Constraint

$\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}\left(\mathrm{𝙽𝙻𝙾𝙾𝙿},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)$

Synonym

$\mathrm{𝚐𝚌𝚌}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$.

Arguments
 $\mathrm{𝙽𝙻𝙾𝙾𝙿}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Restrictions
 $\mathrm{𝙽𝙻𝙾𝙾𝙿}\ge 0$ $\mathrm{𝙽𝙻𝙾𝙾𝙿}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)$ $|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>0$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\left[\mathrm{𝚟𝚊𝚕},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\right]\right)$ $\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}$$\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)$ $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\ge 0$ $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$
Purpose

$\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}$ $\left(1\le i\le |\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\right)$ is equal to the number of variables $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[j\right].\mathrm{𝚟𝚊𝚛}$ $\left(j\ne i,1\le j\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right)$ that are assigned value $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}$.

The number of assignments of the form $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}=i$ ($i\in \left[1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right]$) is equal to $\mathrm{𝙽𝙻𝙾𝙾𝙿}$.

Example
$\left(\begin{array}{c}1,〈1,1,8,6〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚟𝚊𝚕}-1\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1,\hfill \\ \mathrm{𝚟𝚊𝚕}-5\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-0,\hfill \\ \mathrm{𝚟𝚊𝚕}-6\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1\hfill \end{array}〉\hfill \end{array}\right)$

The $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ constraint holds since:

• Values 1, 5 and 6 are respectively assigned to the set of variables $\left\{\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[2\right].\mathrm{𝚟𝚊𝚛}\right\}$ (i.e., 1 occurrence of value 1), $\left\{\right\}$ (i.e., no occurrence of value 5) and $\left\{\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[4\right].\mathrm{𝚟𝚊𝚛}\right\}$ (i.e., 1 occurrence of value 6). Note that, due to the definition of the constraint, the fact that $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[1\right].\mathrm{𝚟𝚊𝚛}$ is assigned to 1 is not counted.

• In addition the number of assignments of the form $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}=i$ ($i\in \left[1,4\right]$) is equal to $\mathrm{𝙽𝙻𝙾𝙾𝙿}=1$.

Typical
 $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1$ $\mathrm{𝚛𝚊𝚗𝚐𝚎}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1$ $|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>1$ $\mathrm{𝚛𝚊𝚗𝚐𝚎}$$\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\right)>1$ $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|$
Symmetry

Items of $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ are permutable.

Usage

Within the context of the $\mathrm{𝚝𝚛𝚎𝚎}$ constraint the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ constraint allows to model a minimum and maximum degree constraint on each vertex of our trees.

Algorithm

The flow algorithm that handles the original $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ constraint [Regin96] can be adapted to the context of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ constraint. This is done by creating an extra value node representing the loops corresponding to the roots of the trees.

See also

related: $\mathrm{𝚝𝚛𝚎𝚎}$ (graph partitioning by a set of trees with degree restrictions).

root concept: $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (assignment of a $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ to its position is ignored).

specialisation: $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ ($\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ replaced by $\mathrm{𝚏𝚒𝚡𝚎𝚍}$ $\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$).

Keywords

For all items of $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$:

Arc input(s)

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$

Arc generator
$\mathrm{𝑆𝐸𝐿𝐹}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)$

Arc arity
Arc constraint(s)
 $•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$ $•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚔𝚎𝚢}\ne \mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$
Graph property(ies)
$\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}$$=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}$

Arc input(s)

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$

Arc generator
$\mathrm{𝑆𝐸𝐿𝐹}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚔𝚎𝚢}$
Graph property(ies)
$\mathrm{𝐍𝐀𝐑𝐂}$$=\mathrm{𝙽𝙻𝙾𝙾𝙿}$

Graph model

Since, within the context of the first graph constraint, we want to express one unary constraint for each value we use the “For all items of $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$” iterator. Part (A) of Figure 5.141.1 shows the initial graphs associated with each value 1, 5 and 6 of the $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ collection of the Example slot. Part (B) of Figure 5.141.1 shows the two corresponding final graphs respectively associated with values 1 and 6 that are both assigned to the variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection (since value 5 is not assigned to any variable of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection the final graph associated with value 5 is empty). Since we use the $\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}$ graph property, the vertices of the final graphs are stressed in bold.

##### Figure 5.141.1. Initial and final graph of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ constraint  (a) (b)