## 5.357. soft_all_equal_min_ctr

Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi ½},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Synonyms

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

Arguments
 $\mathrm{\pi ½}$ $\mathrm{\pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ½}\beta ₯0$ $\mathrm{\pi ½}\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|*|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $\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)$
Purpose

Consider the equality constraints involving two distinct variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$. Among the previous set of constraints, $\mathrm{\pi ½}$ is less than or equal to the number of equality constraints that hold.

Example
$\left(6,β©5,1,5,5βͺ\right)$

Within the collection $\beta ©5,1,5,5\beta ͺ$ six equality constraints holds. Consequently, the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ constraint holds since the argument $\mathrm{\pi ½}=6$ is less than or equal to the number of equality constraints that hold.

Typical
 $\mathrm{\pi ½}>0$ $\mathrm{\pi ½}<|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|*|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$
Symmetries
• $\mathrm{\pi ½}$ can be decreased to any value $\beta ₯0$.

• 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.

Remark

It was shown inΒ [HebrardSullivanRazgon08] that, finding out whether the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ constraint has a solution or not is NP-hard. This was achieved by reduction from 3-dimensional-matching. Hebrard et al. also identify a tractable class when no value occurs in more than two variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ that is equivalent to the vertex matching problem. One year later,Β [HebrardMarxSullivanRazgon09] shows how to achieve bound-consistency in polynomial time.

Keywords
Arc input(s)

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

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$

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 }$$\beta ₯\mathrm{\pi ½}$

Graph model

We generate an initial graph with binary equalities constraints between each vertex and its successors. We use the arc generator in order to avoid considering equality constraints between the same variable. The graph property states that $\mathrm{\pi ½}$ is less than or equal to the number of equalities that hold in the final graph.

PartsΒ (A) andΒ (B) of FigureΒ 5.357.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the arcs of the final graph are stressed in bold. Six equality constraints remain in the final graph.