## 5.121. discrepancy

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi Ί}\right)$

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

$\mathrm{\pi Ί}$ is the number of variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ that take their value in their respective sets of bad values.

Example
$\left(\begin{array}{c}β©\begin{array}{cc}\mathrm{\pi \pi \pi }-4\hfill & \mathrm{\pi \pi \pi }-\left\{1,4,6\right\},\hfill \\ \mathrm{\pi \pi \pi }-5\hfill & \mathrm{\pi \pi \pi }-\left\{0,1\right\},\hfill \\ \mathrm{\pi \pi \pi }-5\hfill & \mathrm{\pi \pi \pi }-\left\{1,6,9\right\},\hfill \\ \mathrm{\pi \pi \pi }-4\hfill & \mathrm{\pi \pi \pi }-\left\{1,4\right\},\hfill \\ \mathrm{\pi \pi \pi }-1\hfill & \mathrm{\pi \pi \pi }-\mathrm{\beta  }\hfill \end{array}βͺ,2\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}$ constraint holds since exactly $\mathrm{\pi Ί}=2$ variables (i.e.,Β the first and fourth variables) of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection take their value within their respective sets of bad values.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi Ί}<|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

• All occurrences of two distinct values in $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ or $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be swapped; all occurrences of a value in $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ or $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be renamed to any unused value.

Arg. properties
• Functional dependency: $\mathrm{\pi Ί}$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

• Aggregate: $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left(\mathrm{\pi \pi \pi \pi \pi }\right)$, $\mathrm{\pi Ί}\left(+\right)$.

Remark

Limited discrepancy search was first introduced by M.Β L.Β Ginsberg and W.Β D.Β Harvey as a search technique inΒ [GinsbergHarvey95]. Later on, discrepancy based filtering was presented in the PhD thesis of F.Β FocacciΒ [Focacci01]. Finally the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}$ constraint was explicitly defined in the PhD thesis of W.-J.Β vanΒ HoeveΒ [vanHoeve05].

Keywords
Arc input(s)

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

Arc generator
$\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi Ί}$

Graph model

The arc constraint corresponds to the constraint $\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }\right)$ defined in this catalogue. We employ the $\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$ arc generator in order to produce an initial graph with a single loop on each vertex.

PartsΒ (A) andΒ (B) of FigureΒ 5.121.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the loops of the final graph are stressed in bold.