5.413. used_by_interval

 DESCRIPTION LINKS GRAPH
Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\right)$

Arguments
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|\beta ₯|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$ $\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)$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}>0$
Purpose

Let ${N}_{i}$ (respectively ${M}_{i}$) denote the number of variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ (respectively $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$) that take a value in the interval $\left[\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·i,\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·i+\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}-1\right]$. For all integer $i$ we have ${M}_{i}>0\beta {N}_{i}\beta ₯{M}_{i}$.

Example
$\left(β©1,9,1,8,6,2βͺ,β©1,0,7,7βͺ,3\right)$

In the example, the third argument $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}=3$ defines the following family of intervals $\left[3Β·k,3Β·k+2\right]$, where $k$ is an integer. Consequently the values of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}=\beta ©1,0,7,7\beta ͺ$ are respectively located within intervals $\left[0,2\right]$, $\left[0,2\right]$, $\left[6,8\right]$, $\left[6,8\right]$. Therefore intervals $\left[0,2\right]$ and $\left[6,8\right]$ are respectively used 2 and 2 times.

Similarly, the values of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}=\beta ©1,9,1,8,6,2\beta ͺ$ are respectively located within intervals $\left[0,2\right]$, $\left[9,11\right]$, $\left[0,2\right]$, $\left[6,8\right]$, $\left[6,8\right]$, $\left[0,2\right]$. Therefore intervals $\left[0,2\right]$, $\left[6,8\right]$ and $\left[9,11\right]$ are respectively used 3, 2 and 1 times.

Consequently, the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since, for each interval associated with the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}=\beta ©1,0,7,7\beta ͺ$, its number of occurrences within $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}=\beta ©1,9,1,8,6,2\beta ͺ$ is greater than or equal to its number of occurrences within $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$:

• Interval $\left[0,2\right]$ occurs 3 times within $\beta ©1,9,1,8,6,2\beta ͺ$ and 2 times within $\beta ©1,0,7,7\beta ͺ$.

• Interval $\left[6,8\right]$ occurs 2 times within $\beta ©1,9,1,8,6,2\beta ͺ$ and 2 times within $\beta ©1,0,7,7\beta ͺ$.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }\right)>1$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}>1$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}<$$\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}<$$\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Symmetries
• 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.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ that belongs to the $k$-th interval, of size $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$, can be replaced by any other value of the same interval.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ that belongs to the $k$-th interval, of size $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$, can be replaced by any other value of the same interval.

Arg. properties
• Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$.

• Extensible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$.

• Aggregate: $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}\left(\mathrm{\pi \pi \pi \pi \pi }\right)$, $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\left(\mathrm{\pi \pi \pi \pi \pi }\right)$, $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\left(\mathrm{\pi \pi }\right)$.

Reformulation

The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\beta ©\mathrm{\pi \pi \pi }-{U}_{1}\mathrm{\pi \pi \pi }-{U}_{2},\beta ―,\mathrm{\pi \pi \pi }-{U}_{|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|}\beta ͺ,\beta ©\mathrm{\pi \pi \pi }-{V}_{1}\mathrm{\pi \pi \pi }-{V}_{2},\beta ―,\mathrm{\pi \pi \pi }-{V}_{|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|}\beta ͺ,\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\right)$ constraint can be expressed by introducing $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|+|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$ quotient variables

Β Β Β ${U}_{i}=\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·{P}_{i}+{R}_{i}$, ${R}_{i}\beta \left[0,\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}-1\right]$ $\left(i\beta \left[1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|\right]\right)$,

Β Β Β ${V}_{i}=\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·{Q}_{i}+{S}_{i}$, ${S}_{i}\beta \left[0,\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}-1\right]$ $\left(i\beta \left[1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|\right]\right)$,

in term of a conjunction of $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$ reified constraints of the form:

Β Β Β ${\beta }_{1\beta €j\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|}\left({Q}_{i}={P}_{j}\right)\beta ₯{\beta }_{1\beta €j\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|}\left({Q}_{i}={Q}_{j}\right)$ $\left(i\beta \left[1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|\right]\right)$.

Used in
See also

specialisation: $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }/\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$).

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)
$\begin{array}{c}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}=\hfill \\ \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\hfill \end{array}$
Graph property(ies)
 $\beta ’\text{for}\text{all}\text{connected}\text{components:}$$\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\beta ₯$$\mathrm{\pi \pi \pi \pi \pi }$ $\beta ’$$\mathrm{\pi \pi \pi \pi \pi }$$=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.413.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ and $\mathrm{\pi \pi \pi \pi \pi }$ graph properties, the source and sink vertices of the final graph are stressed with a double circle. Since there is a constraint on each connected component of the final graph we also show the different connected components. Each of them corresponds to an equivalence class according to the arc constraint. Note that the vertex corresponding to the variable that takes value 9 was removed from the final graph since there is no arc for which the associated equivalence constraint holds. The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since:

• For each connected component of the final graph the number of sources is greater than or equal to the number of sinks.

• The number of sinks of the final graph is equal to $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$.

Signature

Since the initial graph contains only sources and sinks, and since sources of the initial graph cannot become sinks of the final graph, we have that the maximum number of sinks of the final graph is equal to $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }2|$. Therefore we can rewrite $\mathrm{\pi \pi \pi \pi \pi }=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }2|$ to $\mathrm{\pi \pi \pi \pi \pi }\beta ₯|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }2|$ and simplify $\underset{Μ²}{\stackrel{Β―}{\mathrm{\pi \pi \pi \pi \pi }}}$ to $\stackrel{Β―}{\mathrm{\pi \pi \pi \pi \pi }}$.