5.370. some_equal

Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Synonyms

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$.

Argument
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$
Purpose

Enforce at least two variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ to be assigned the same value.

Example
$\left(β©1,4,1,6βͺ\right)$

The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since the first and the third variables are both assigned the same value 1.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>2$ $\mathrm{\pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>2$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

• All occurrences of two distinct values of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be swapped; all occurrences of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be renamed to any unused value.

Arg. properties

Extensible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

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

Number of solutions for $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$: domains $0..n$

Used in
Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\left(<\right)\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$>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.

PartsΒ (A) andΒ (B) of FigureΒ 5.370.1 respectively show the initial and final graph associated with the Example slot. The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since the final graph has at one arc, i.e.Β two variables are assigned the same value.