## 5.109. dag

Origin
Constraint

$\mathrm{𝚍𝚊𝚐}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)$

Argument
 $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)$ $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1$ $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|$ $\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}$$\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)$ $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1$ $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|$
Purpose

Consider a digraph $G$ described by the $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ collection. Select a subset of arcs of $G$ so that the corresponding graph does not contain any circuit.

Example
$\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{3,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing ,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing ,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{6\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-6\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing \hfill \end{array}〉\hfill \end{array}\right)$

The $\mathrm{𝚍𝚊𝚐}$ constraint holds since the $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ collection depicts a graph without circuit.

Typical
$|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2$
Symmetry

Items of $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ are permutable.

Algorithm

A filtering algorithm for the $\mathrm{𝚍𝚊𝚐}$ constraint is given in [Dooms06]. It removes potential arcs that would create a circuit of mandatory arcs.

Keywords
Arc input(s)

$\mathrm{𝙽𝙾𝙳𝙴𝚂}$

Arc generator
$\mathrm{𝑆𝐸𝐿𝐹}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}$$\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}.\mathrm{𝚔𝚎𝚢},\mathrm{𝚗𝚘𝚍𝚎𝚜}.\mathrm{𝚜𝚞𝚌𝚌}\right)$
Graph property(ies)
$\mathrm{𝐍𝐀𝐑𝐂}$$=0$

Arc input(s)

$\mathrm{𝙽𝙾𝙳𝙴𝚂}$

Arc generator
$\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}$$\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\right)$
Graph property(ies)
$\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}$$\le 1$

Graph model

The first graph constraint removes the loop of each vertex. The second graph constraint forbids the creation of circuits involving more than one vertex.

Part (A) of Figure 5.109.1 shows the initial graph associated with the second graph constraint of the Example slot. This initial graph from which we start is derived from the set associated with each vertex. Each set describes the potential values of the $\mathrm{𝚜𝚞𝚌𝚌}$ attribute of a given vertex. Part (B) of Figure 5.109.1 gives the final graph associated with the Example slot.

##### Figure 5.109.1. Initial and final graph of the $\mathrm{𝚍𝚊𝚐}$ set constraint  (a) (b)