## 5.207. k_same

Origin
Constraint

$𝚔_\mathrm{𝚜𝚊𝚖𝚎}\left(\mathrm{𝚂𝙴𝚃𝚂}\right)$

Type
 $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Argument
 $\mathrm{𝚂𝙴𝚃𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚜𝚎𝚝}-\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)$ $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\ge 1$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚂𝙴𝚃𝚂},\mathrm{𝚜𝚎𝚝}\right)$ $|\mathrm{𝚂𝙴𝚃𝚂}|>1$ $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}$$\left(\mathrm{𝚂𝙴𝚃𝚂},\mathrm{𝚜𝚎𝚝}\right)$
Purpose

Given $|\mathrm{𝚂𝙴𝚃𝚂}|$ sets, each containing the same number of domain variables, the $𝚔_\mathrm{𝚜𝚊𝚖𝚎}$ constraint forces that the multisets of values assigned to each set are all identical.

Example
$\left(\begin{array}{c}〈\begin{array}{c}\mathrm{𝚜𝚎𝚝}-〈1,9,1,5,2,1〉,\hfill \\ \mathrm{𝚜𝚎𝚝}-〈9,1,1,1,2,5〉,\hfill \\ \mathrm{𝚜𝚎𝚝}-〈5,2,1,1,9,1〉\hfill \end{array}〉\hfill \end{array}\right)$

The $𝚔_\mathrm{𝚜𝚊𝚖𝚎}$ constraint holds since:

• The first and second collections of variables are assigned to the same multiset.

• The second and third collections of variables are also assigned to the same multiset.

Typical
$|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1$
Symmetries
• Items of $\mathrm{𝚂𝙴𝚃𝚂}$ are permutable.

• Items of $\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}$ are permutable.

• All occurrences of two distinct values of $\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}.\mathrm{𝚟𝚊𝚛}$ can be swapped; all occurrences of a value of $\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}.\mathrm{𝚟𝚊𝚛}$ can be renamed to any unused value.

Arg. properties

Contractible wrt. $\mathrm{𝚂𝙴𝚃𝚂}$.

Remark

It was shown in [ElbassioniKatrielKutzMahajan05] that, finding out whether the $𝚔_\mathrm{𝚜𝚊𝚖𝚎}$ constraint has a solution or not is NP-hard when we have more than one $\mathrm{𝚜𝚊𝚖𝚎}$ constraint. This was achieved by reduction from 3-dimensional-matching in the context where we have 2 $\mathrm{𝚜𝚊𝚖𝚎}$ constraints.

Keywords
Arc input(s)

$\mathrm{𝚂𝙴𝚃𝚂}$

Arc generator
$\mathrm{𝑃𝐴𝑇𝐻}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚜𝚎𝚝}\mathtt{1},\mathrm{𝚜𝚎𝚝}\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚜𝚊𝚖𝚎}$$\left(\mathrm{𝚜𝚎𝚝}\mathtt{1}.\mathrm{𝚜𝚎𝚝},\mathrm{𝚜𝚎𝚝}\mathtt{2}.\mathrm{𝚜𝚎𝚝}\right)$
Graph property(ies)
$\mathrm{𝐍𝐀𝐑𝐂}$$=|\mathrm{𝚂𝙴𝚃𝚂}|-1$

Graph model

Parts (A) and (B) of Figure 5.207.1 respectively show the initial and final graph associated with the Example slot. To each vertex corresponds a collection of variables, while to each arc corresponds a $\mathrm{𝚜𝚊𝚖𝚎}$ constraint.

##### Figure 5.207.1. Initial and final graph of the $𝚔_\mathrm{𝚜𝚊𝚖𝚎}$ constraint  (a) (b)