5.365. soft_same_var

Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi ²},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\right)$

Synonym

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$.

Arguments
 $\mathrm{\pi ²}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ²}\beta ₯0$ $\mathrm{\pi ²}\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi \pi \pi }\right)$
Purpose

$\mathrm{\pi ²}$ is the minimum number of values to change in the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ collections so that the variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ collection correspond to the variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ collection according to a permutation.

Example
$\left(4,β©9,9,9,9,9,1βͺ,β©9,1,1,1,1,8βͺ\right)$

As illustrated by FigureΒ 5.365.1, there is a correspondence between two pairs of values of the collections $\beta ©9,9,9,9,9,1\beta ͺ$ and $\beta ©9,1,1,1,1,8\beta ͺ$. Consequently, we must unset at least $6-2$ items (6 is the number of items of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ collections). The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi ²}$ is set to $6-2$.

Typical
 $\mathrm{\pi ²}>0$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)>1$
Symmetries
• Arguments are permutable w.r.t. permutation $\left(\mathrm{\pi ²}\right)$ $\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\right)$.

• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ are permutable.

• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ are permutable.

• All occurrences of two distinct values in $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ or $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ can be swapped; all occurrences of a value in $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ or $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ can be renamed to any unused value.

Usage
Algorithm

A first filtering algorithm is described inΒ [vanHoeve05]. A second filtering algorithm is presented inΒ [Cymer12], [CymerPhD13].

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$

Arc generator
$\mathrm{\pi \pi  \pi \pi ·\pi \pi Ά\pi }$$\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 \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|-\mathrm{\pi ²}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.365.2 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ graph property, the source and sink vertices of the final graph are stressed with a double circle. The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ constraint holds since the cost 4 corresponds to the difference between the number of variables of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and the sum over the different connected components of the minimum number of sources and sinks.