5.333. roots

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi }\left(\mathrm{\pi },\mathrm{\pi },\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

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

$\mathrm{\pi }$ is the set of indices of the variables in the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ taking their values in $\mathrm{\pi }$; $\mathrm{\pi }=\left\{i|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }\beta \mathrm{\pi }\right\}$. Positions are numbered from 1.

Example
$\left(\left\{2,4,5\right\},\left\{2,3,8\right\},β©1,3,1,2,3βͺ\right)$

The $\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since values 2 and 3 in $\mathrm{\pi }$ occur in the collection $\beta ©1,3,1,2,3\beta ͺ$ only at positions $\mathrm{\pi }=\left\{2,4,5\right\}$. The value $8\beta \mathrm{\pi }$ does not occur within the collection $\beta ©1,3,1,2,3\beta ͺ$.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$
Usage

BessiΓ¨re et al. showedΒ [BessiereHebrardHnichKiziltanWalsh05IJCAI] that many counting and occurence constraints can be specified with two global primitives: $\mathrm{\pi \pi \pi \pi \pi }$ and $\mathrm{\pi \pi \pi \pi \pi }$. For instance, the $\mathrm{\pi \pi \pi \pi \pi }$ constraint can be decomposed into one $\mathrm{\pi \pi \pi \pi \pi }$ constraint: $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi »},\mathrm{\pi  \pi °\pi \pi },\mathrm{\pi Ύ\pi Ώ},\mathrm{\pi ½\pi  \pi °\pi }\right)$ iff $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi },\left\{\mathrm{\pi  \pi °\pi »}\right\},\mathrm{\pi  \pi °\pi \pi }\right)\beta §|\mathrm{\pi }|\mathrm{\pi Ύ\pi Ώ}\mathrm{\pi ½\pi  \pi °\pi }$.

$\mathrm{\pi \pi \pi \pi \pi }$ does not count but collects the set of variables using particular values. It provides then a way of channeling. $\mathrm{\pi \pi \pi \pi \pi }$ generalises, for instance, the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint, $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi },\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }\right)$ iff $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi },\left\{1\right\},\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }.\mathrm{\pi \pi \pi \pi }\right)$, or may be used instead of the $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$.

Other examples of reformulations are given inΒ [BessiereHebrardHnichKiziltanWalsh09].

Algorithm

InΒ [BessiereHebrardHnichKiziltanWalsh06CP], BessiΓ¨re et al. shows that enforcing hybrid-consistency on $\mathrm{\pi \pi \pi \pi \pi }$ is NP-hard. They consider the decomposition of $\mathrm{\pi \pi \pi \pi \pi }$ into a network of ternary constraints: $\beta i$, $i\beta \mathrm{\pi }\beta \mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }\beta \mathrm{\pi }$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }\beta \mathrm{\pi }\beta §i\beta \mathrm{\pi }$. Enforcing bound consistency on the decomposition achieves bound consistency on $\mathrm{\pi \pi \pi \pi \pi }$. Enforcing hybrid consistency on the decomposition achieves at least bound consistency on $\mathrm{\pi \pi \pi \pi \pi }$, until hybrid consistency in some special cases:

• $\mathrm{\pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }\right)\beta \underset{Μ²}{\mathrm{\pi }},\beta i\beta \underset{Μ²}{\mathrm{\pi }}$,

• $\mathrm{\pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }\right)\beta ©\stackrel{Β―}{\mathrm{\pi }}=\mathrm{\beta  },\beta i\beta \stackrel{Β―}{\mathrm{\pi }}$,

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

• $\mathrm{\pi }$ is ground.

Enforcing hybrid consistency on the decomposition can be done in $O\left(nd\right)$ with $n=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ and $d$ the maximum domain size of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }$ and $\mathrm{\pi }$.

Systems

roots in Gecode, roots in MiniZinc.

Keywords
Derived Collection
$\mathrm{\pi \pi \pi }\left(\mathrm{\pi \pi ΄\pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi }-\mathrm{\pi \pi \pi \pi },\mathrm{\pi }-\mathrm{\pi \pi \pi \pi }\right),\left[\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi }-\mathrm{\pi },\mathrm{\pi }-\mathrm{\pi }\right)\right]\right)$
Arc input(s)

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

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 },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
$\begin{array}{c}\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi ’},\mathrm{\pi \pi \pi \pi }.\mathrm{\pi }\right)\beta \hfill \\ \mathrm{\pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi }.\mathrm{\pi }\right)\hfill \end{array}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$

Graph model