## 5.330. symmetric

Origin
Constraint

$\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }\right)$

Argument
 $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi },\left[\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi }\right]\right)$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta ₯1$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta €|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi },\mathrm{\pi \pi \pi \pi \pi ‘}\right)$
Purpose

Consider a digraph $G$ described by the $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ collection. Select a subset of arcs of $G$ so that the corresponding graph is symmetric (i.e.,Β if there is an arc from $i$ to $j$, there is also an arc from $j$ to $i$).

Example
$\left(\begin{array}{c}β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi \pi ‘}-1\hfill & \mathrm{\pi \pi \pi \pi }-\left\{1,2,3\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-2\hfill & \mathrm{\pi \pi \pi \pi }-\left\{1,3\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-3\hfill & \mathrm{\pi \pi \pi \pi }-\left\{1,2\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-4\hfill & \mathrm{\pi \pi \pi \pi }-\left\{5,6\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-5\hfill & \mathrm{\pi \pi \pi \pi }-\left\{4\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-6\hfill & \mathrm{\pi \pi \pi \pi }-\left\{4\right\}\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since the $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ collection depicts a symmetric graph.

Symmetry

Items of $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ are permutable.

Algorithm

The filtering algorithm for the $\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }$ constraint is given inΒ [Dooms06]. It removes (respectively imposes) the arcs $\left(i,j\right)$ for which the arc $\left(j,i\right)$ is not present (respectively is present). It has an overall complexity of $O\left(n+m\right)$ where $n$ and $m$ respectively denote the number of vertices and the number of arcs of the initial graph.

Keywords
Arc input(s)

$\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi }\right)$
Graph class
$\mathrm{\pi \pi \pi Ό\pi Ό\pi ΄\pi \pi \pi Έ\pi ²}$

Graph model

PartΒ (A) of FigureΒ 5.330.1 shows the initial graph from which we start. It is derived from the set associated with each vertex. Each set describes the potential values of the $\mathrm{\pi \pi \pi \pi }$ attribute of a given vertex. PartΒ (B) of FigureΒ 5.330.1 gives the final graph associated with the Example slot.