Used for defining $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}\_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}$.

$\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂})$

$\mathrm{𝚐𝚌𝚌}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$, $\mathrm{𝚐𝚌𝚌}$.

The $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint holds since values 3, 5 and 6 are respectively used 2 ($2\le 2\le 3$), 0 ($0\le 0\le 1$) and 1 ($1\le 1\le 2$) times within the collection $\langle 3,3,8,6\rangle$ and since no constraint was specified for value 8.

• Items of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ are permutable.

• An occurrence of a value of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}$ that does not belong to $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$ can be replaced by any other value that also does not belong to $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$.

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

• $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}$ can be decreased to any value $\ge 0$.

• $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}$ can be increased to any value $\le \left|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right|$.

• All occurrences of two distinct values in $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}$ or $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$ can be swapped; all occurrences of a value in $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}$ or $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$ can be renamed to any unused value.

Contractible wrt. $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$.

Within the context of linear programming [Hooker07book] provides relaxations of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint.

In MiniZinc (http://www.minizinc.org/) there is also a $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}\_\mathrm{𝚌𝚕𝚘𝚜𝚎𝚍}$ constraint where all variables must be assigned a value from the $\mathrm{𝚟𝚊𝚕}$ attribute.

A filtering algorithm achieving arc-consistency for the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint is given in [Regin96]. This algorithm is based on a flow model of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint where there is a one-to-one correspondence between feasible flows in the flow model and solutions of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint. The leftmost part of Figure 3.7.29 illustrates this flow model.

The $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint is entailed if and only if for each value $v$ equal to $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}$ (with $1\le i\le \left|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right|$) the following two conditions hold:

1. The number of variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection assigned value $v$ is greater than or equal to $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚒𝚗}$.

2. The number of variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection that can potentially be assigned value $v$ is less than or equal to $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚊𝚡}$.

A reformulation of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$, involving linear constraints, preserving bound-consistency was introduced in [BessiereKatsirelosNarodytskaQuimperWalsh09IJCAI]. For each potential interval $[l,u]$ of consecutive values this model uses $\left|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right|$ 0-1 variables $B_{1,l,u},B_{2,l,u},\cdots ,B_{\left|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right|,l,u}$ for modelling the fact that each variable of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ is assigned a value within interval $[l,u]$ (i.e., $\forall i\in [1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left|\right]:B_{i,l,u}\iff l\le \mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}\wedge \mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}\le u$), as well as one domain variable $C_{l,u}$ for counting how many values of $[l,u]$ are assigned to variables of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ (i.e. $C_{l,u}=B_{1,l,u}+B_{2,l,u}+\cdots +B_{\left|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right|,l,u}$). The lower and upper bounds of variable $C_{l,u}$ are respectively initially set with respect to the minimum and maximum number of possible occurrences of the values of interval $[l,u]$. Finally, assuming that $s$ is the smallest value that can be assigned to the variables of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$, the constraint $C_{s,u}=C_{s,k}+C_{k+1,u}$ is stated for each $k\in [s,u-1]$.

globalCardinality in Choco, global_cardinality_low_up in MiniZinc.

$\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}\_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}$.

common keyword: $\mathrm{𝚘𝚙𝚎𝚗}\_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (assignment,counting constraint).

generalisation: $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ ($\mathrm{𝚏𝚒𝚡𝚎𝚍}$ $\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ replaced by $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$).

implied by: $\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}\_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (a $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint where the $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}$ are increasing), $\mathrm{𝚜𝚊𝚖𝚎}\_\mathrm{𝚊𝚗𝚍}\_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$.

related: $\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}\_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (restrictions are done on nested sets of values, all starting from first value).

shift of concept: $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}\_\mathrm{𝚗𝚘}\_\mathrm{𝚕𝚘𝚘𝚙}$ (assignment of a $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ to its position is ignored).

soft variant: $\mathrm{𝚘𝚙𝚎𝚗}\_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ (a $\mathrm{𝚜𝚎𝚝}$ $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ defines the set of variables that are actually considered).

specialisation: $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (each value should occur at most once).

system of constraints: $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}\_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}$ (one $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚕𝚘𝚠}\_\mathrm{𝚞𝚙}$ constraint for each sliding sequence of $\mathrm{𝚂𝙴𝚀}$ consecutive $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}$).

application area: assignment.

constraint type: value constraint, counting constraint.

filtering: flow, arc-consistency, bound-consistency, DFS-bottleneck, entailment.

For all items of $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$: