## 5.268. multi_inter_distance

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi },\mathrm{\pi ³\pi Έ\pi \pi }\right)$

Synonyms

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

Arguments
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$ $\mathrm{\pi \pi \pi }$ $\mathrm{\pi ³\pi Έ\pi \pi }$ $\mathrm{\pi \pi \pi }$
Restrictions
 $\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)$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }>0$ $\mathrm{\pi ³\pi Έ\pi \pi }>0$
Purpose

Enforce that at most $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$ variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are assigned values in any set consisting of $\mathrm{\pi ³\pi Έ\pi \pi }$ consecutive integer values.

Example
$\left(β©4,0,9,4,7βͺ,2,3\right)$

The $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since, for each set of $\mathrm{\pi ³\pi Έ\pi \pi }=3$ consecutive values, no more than $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }=2$ variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection $\beta ©4,0,9,4,7\beta ͺ$ are assigned a value from that set:

• At most two, in fact one, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{0,1,2\right\}$.

• At most two, in fact zero, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{1,2,3\right\}$.

• At most two, in fact two, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{2,3,4\right\}$.

• At most two, in fact two, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{3,4,5\right\}$.

• At most two, in fact two, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{4,5,6\right\}$.

• At most two, in fact one, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{5,6,7\right\}$.

• At most two, in fact one, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{6,7,8\right\}$.

• At most two, in fact two, variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned a value from the set $\left\{7,8,9\right\}$.

Typical
 $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }>1$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }<|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $\mathrm{\pi ³\pi Έ\pi \pi }>1$ $\mathrm{\pi ³\pi Έ\pi \pi }<$$\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

• One and the same constant can be added to the $\mathrm{\pi \pi \pi }$ attribute of all items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

• $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$ can be increased.

• $\mathrm{\pi Ό\pi Έ\pi ½\pi ³\pi Έ\pi \pi }$ can be decreased to any value $\beta ₯1$.

Arg. properties

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

Usage

The $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint was tested for scheduling tasks that all have the same fixed duration in the context of air traffic management.

Algorithm

P.Β Ouellet and C.-G.Β Quimper came up with a cubic time complexity algorithm achieving bound-consistency inΒ [OuelletQuimper11].

generalisation: $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi }\mathrm{\pi \pi \pi \pi \pi \pi \pi }$, of same length, replaced by $\mathrm{\pi \pi \pi \pi }\mathrm{\pi \pi \pi \pi \pi \pi \pi }$).
specialisation: $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$Β ($\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$ parameter set to 1), $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi }$Β (window of $\mathrm{\pi ³\pi Έ\pi \pi }$ consecutive values replaced by window of size 1).