## 5.178. k_same

Origin
Constraint

$\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi ΄\pi \pi }\right)$

Type
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Argument
 $\mathrm{\pi \pi ΄\pi \pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>0$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi ΄\pi \pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi \pi ΄\pi \pi }|>1$ $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi £\pi }$$\left(\mathrm{\pi \pi ΄\pi \pi },\mathrm{\pi \pi \pi }\right)$
Purpose

Given $|\mathrm{\pi \pi ΄\pi \pi }|$ sets, each containing the same number of domain variables, the $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }$ constraint enforces that the multisets of values assigned to each set are all identical.

Example
$\left(\begin{array}{c}β©\begin{array}{c}\mathrm{\pi \pi \pi }-β©\begin{array}{c}\mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-9,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-5,\hfill \\ \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-1\hfill \end{array}βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©\begin{array}{c}\mathrm{\pi \pi \pi }-9,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-5\hfill \end{array}βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©\begin{array}{c}\mathrm{\pi \pi \pi }-5,\hfill \\ \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-9,\hfill \\ \mathrm{\pi \pi \pi }-1\hfill \end{array}βͺ\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }$ 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
$|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$
Symmetries
• Items of $\mathrm{\pi \pi ΄\pi \pi }$ are permutable.

• Items of $\mathrm{\pi \pi ΄\pi \pi }.\mathrm{\pi \pi \pi }$ are permutable.

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

Remark

It was shown inΒ [ElbassioniKatrielKutzMahajan05] that, finding out whether the $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }$ constraint has a solution or not is NP-hard when we have more than one $\mathrm{\pi \pi \pi \pi }$ constraint. This was achieved by reduction from 3-dimensional-matching in the context where we have 2 $\mathrm{\pi \pi \pi \pi }$ constraints.

Keywords
Arc input(s)

$\mathrm{\pi \pi ΄\pi \pi }$

Arc generator
$\mathrm{\pi \pi ΄\pi \pi »}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi \pi ΄\pi \pi }|-1$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.178.1 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 $\mathrm{\pi \pi \pi \pi }$ constraint.