## 5.370. some_equal

Origin
Constraint

$\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚𝚞𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)$

Synonyms

$\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚}$, $\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$, $\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}$, $\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}$, $\mathrm{𝚗𝚘𝚝}_\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}$.

Argument
 $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)$ $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1$
Purpose

Enforce at least two variables of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ to be assigned the same value.

Example
$\left(〈1,4,1,6〉\right)$

The $\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ constraint holds since the first and the third variables are both assigned the same value 1.

Typical
 $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>2$ $\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>2$
Symmetries
• Items of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ are permutable.

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

Arg. properties

Extensible wrt. $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$.

Counting
 Length ($n$) 2 3 4 5 6 7 8 Solutions 3 40 505 7056 112609 2056832 42683841

Number of solutions for $\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚𝚞𝚊𝚕}$: domains $0..n$  Used in
Keywords
Arc input(s)

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$

Arc generator
$\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}$$\left(<\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}$
Graph property(ies)
$\mathrm{𝐍𝐀𝐑𝐂}$$>0$

Graph model

We generate a clique with an equality constraint between each pair of distinct vertices and state that the number of arcs of the final graph should be strictly greater than 0.

Parts (A) and (B) of Figure 5.370.1 respectively show the initial and final graph associated with the Example slot. The $\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ constraint holds since the final graph has at one arc, i.e. two variables are assigned the same value.

##### Figure 5.370.1. Initial and final graph of the $\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ constraint  (a) (b)