## 5.365. soft_same_var

Origin
Constraint

$\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚟𝚊𝚛}\left(𝙲,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}\right)$

Synonym

$\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚜𝚊𝚖𝚎}$.

Arguments
 $𝙲$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Restrictions
 $𝙲\ge 0$ $𝙲\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|$ $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}|$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚟𝚊𝚛}\right)$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2},\mathrm{𝚟𝚊𝚛}\right)$
Purpose

$𝙲$ is the minimum number of values to change in the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ and $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$ collections so that the variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$ collection correspond to the variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ collection according to a permutation.

Example
$\left(4,〈9,9,9,9,9,1〉,〈9,1,1,1,1,8〉\right)$

As illustrated by Figure 5.365.1, there is a correspondence between two pairs of values of the collections $〈9,9,9,9,9,1〉$ and $〈9,1,1,1,1,8〉$. Consequently, we must unset at least $6-2$ items (6 is the number of items of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ and $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$ collections). The $\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚟𝚊𝚛}$ constraint holds since its first argument $𝙲$ is set to $6-2$.

Typical
 $𝙲>0$ $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|>1$ $\mathrm{𝚛𝚊𝚗𝚐𝚎}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}.\mathrm{𝚟𝚊𝚛}\right)>1$ $\mathrm{𝚛𝚊𝚗𝚐𝚎}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right)>1$
Symmetries
• Arguments are permutable w.r.t. permutation $\left(𝙲\right)$ $\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}\right)$.

• Items of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ are permutable.

• Items of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$ are permutable.

• All occurrences of two distinct values in $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}.\mathrm{𝚟𝚊𝚛}$ or $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}.\mathrm{𝚟𝚊𝚛}$ can be swapped; all occurrences of a value in $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}.\mathrm{𝚟𝚊𝚛}$ or $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}.\mathrm{𝚟𝚊𝚛}$ can be renamed to any unused value.

Usage

A soft $\mathrm{𝚜𝚊𝚖𝚎}$ constraint.

Algorithm

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

Keywords
Arc input(s)

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$

Arc generator
$\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}$$↦\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{𝐍𝐒𝐈𝐍𝐊}_\mathrm{𝐍𝐒𝐎𝐔𝐑𝐂𝐄}$$=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|-𝙲$

Graph model

Parts (A) and (B) of Figure 5.365.2 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{𝐍𝐒𝐈𝐍𝐊}_\mathrm{𝐍𝐒𝐎𝐔𝐑𝐂𝐄}$ graph property, the source and sink vertices of the final graph are stressed with a double circle. The $\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚟𝚊𝚛}$ constraint holds since the cost 4 corresponds to the difference between the number of variables of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ and the sum over the different connected components of the minimum number of sources and sinks.