[Thiel04]

$\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}\_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝙲𝙾𝚂𝚃})$

$\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}\_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$, $\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}\_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}$, $\mathrm{𝚠𝚙𝚊}$.

The $\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}\_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}$ constraint holds since:

• No value, except value $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}=0$, is used more than once.

• $\mathrm{𝙲𝙾𝚂𝚃}=8$ is equal to the sum of the weights 2, $-1$ and 7 of the values 1, 2 and 4 assigned to the variables of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}=\langle 4,0,1,2,0,0\rangle$.

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

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

• All occurrences of two distinct values in $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}$ or $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$ that are both different from $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$ can be swapped; all occurrences of a value in $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}$ or $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}$ that is different from $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$ can be renamed to any unused value that is also different from $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$.

Functional dependency: $\mathrm{𝙲𝙾𝚂𝚃}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$.

In his PhD thesis [Thiel04], Sven Thiel describes the following three potential scenarios of the $\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}\_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}$ constraint:

• Given a set of tasks (i.e., the items of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection), assign to each task a resource (i.e., an item of the $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ collection). Except for the resource associated with value $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$, every resource can be used at most once. The cost of a resource is independent from the task to which the resource is assigned. The cost of value $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$ is equal to 0. The total cost $\mathrm{𝙲𝙾𝚂𝚃}$ of an assignment corresponds to the sum of the costs of the resources effectively assigned to the tasks. Finally we impose an upper bound on the total cost.

• Given a set of persons (i.e., the items of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection), select for each person an offer (i.e., an item of the $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ collection). Except for the offer associated with value $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$, every offer should be selected at most once. The profit associated with an offer is independent from the person that selects the offer. The profit of value $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$ is equal to 0. The total benefit $\mathrm{𝙲𝙾𝚂𝚃}$ is equal to the sum of the profits of the offers effectively selected. In addition we impose a lower bound on the total benefit.

• The last scenario deals with an application to an over-constraint problem involving the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint. Allowing some variables to take an "undefined" value is done by setting all weights of all the values different from $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$ to 1. As a consequence all variables assigned to a value different from $\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}$ will have to take distinct values. The $\mathrm{𝙲𝙾𝚂𝚃}$ variable allows to control the number of such variables.

It was shown in [Thiel04] that, finding out whether the $\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}\_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}$ constraint has a solution or not is NP-hard. This was achieved by reduction from subset sum.

A filtering algorithm is given in [Thiel04]. After showing that, deciding whether the $\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}\_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}$ has a solution is NP-complete, [Thiel04] gives the following results of his filtering algorithm with respect to consistency under the 3 scenarios previously described:

• For scenario 1, if there is no restriction of the lower bound of the $\mathrm{𝙲𝙾𝚂𝚃}$ variable, the filtering algorithm achieves arc-consistency for all variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection (but not for the $\mathrm{𝙲𝙾𝚂𝚃}$ variable itself).

• For scenario 2, if there is no restriction of the upper bound of the $\mathrm{𝙲𝙾𝚂𝚃}$ variable, the filtering algorithm achieves arc-consistency for all variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection (but not for the $\mathrm{𝙲𝙾𝚂𝚃}$ variable itself).

• Finally, for scenario 3, the filtering algorithm achieves arc-consistency for all variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection as well as for the $\mathrm{𝙲𝙾𝚂𝚃}$ variable.

attached to cost variant: $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$, $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}\_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}\_\mathtt{0}$.

common keyword: $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚠𝚒𝚝𝚑}\_\mathrm{𝚌𝚘𝚜𝚝𝚜}$ (weighted assignment), $\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}\_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (cost filtering constraint,weighted assignment), $\mathrm{𝚜𝚘𝚏𝚝}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}\_\mathrm{𝚟𝚊𝚛}$ (soft constraint),

$\mathrm{𝚜𝚞𝚖}\_\mathrm{𝚘𝚏}\_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚜}\_\mathrm{𝚘𝚏}\_\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}\_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$ (weighted assignment).

application area: assignment.

characteristic of a constraint: all different, joker value.

complexity: subset sum.

constraint type: soft constraint, relaxation.

filtering: cost filtering constraint.

modelling: functional dependency.

problems: weighted assignment.