## 5.182. k_used_by

Origin
Constraint

$\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}\left(\mathrm{\pi \pi ΄\pi \pi }\right)$

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 }_\mathrm{\pi \pi \pi \pi \pi \pi \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 of domain variables, the $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ constraint enforces a $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ constraint between each pair of consecutive sets.

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

The $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ constraint holds since:

• The multiset of values $\left\{\left\{1,1,1,2,5,9\right\}\right\}$ associated with the second collection of variables is included into the multiset $\left\{\left\{1,1,1,2,5,9\right\}\right\}$ associated with the first collection of variables.

• The multiset of values $\left\{\left\{1,1,2,5\right\}\right\}$ associated with the third collection of variables is included into the multiset $\left\{\left\{1,1,1,2,5,9\right\}\right\}$ associated with the second collection of variables.

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

Similarly to the $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }$ constraintΒ [ElbassioniKatrielKutzMahajan05], finding out whether the $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ constraint has a solution or not is NP-hard when we have more than one $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ constraint.

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 }_\mathrm{\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 5.182.1 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 $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ constraint.