3.7.101.1. Flow models for $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$, $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$

Figure 3.7.22 presents flow models for the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ and the $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraints. Blue arcs represent feasible flows respectively corresponding to the solutions $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$$\left(〈{x}_{1}=1,{x}_{2}=2,{x}_{3}=3,{x}_{4}=4,{x}_{5}=5〉\right)$ and $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$$\left(\left\{1,2,3,5\right\},〈{x}_{1}=1,{x}_{2}=2,{x}_{3}=3,{x}_{4}=3,{x}_{5}=4〉\right)$, while pink arcs correspond to arcs that cannot carry any flow if the constraint has a solution:

• Within the context of the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint the assignments ${x}_{3}=1$, ${x}_{3}=2$ and ${x}_{4}=2$ are forbidden since values 1 and 2 must be already assigned to ${x}_{1}$ and ${x}_{2}$. Finally the assignments ${x}_{4}=3$ and ${x}_{5}=3$ are also forbidden since values 1, 2 and 3 must be assigned to ${x}_{1}$, ${x}_{2}$ and ${x}_{3}$.

• Note that, within the context of the $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint, the assignment ${x}_{4}=3$ does not matter at all since the position of ${x}_{4}$ within $〈{x}_{1}=1,{x}_{2}=2,{x}_{3}=3,{x}_{4}=3,{x}_{5}=4〉$ does not belong to the set $\left\{1,2,3,5\right\}$. We can only prune according to those variables that for sure should be assigned distinct values. Consequently ${x}_{3}=1$ and ${x}_{3}=2$ are forbidden since values 1 and 2 must already be assigned to ${x}_{1}$ and ${x}_{2}$. Finally the assignment ${x}_{5}=3$ is also forbidden since values 1, 2 and 3 must be assigned to ${x}_{1}$, ${x}_{2}$ and ${x}_{3}$.

Table 3.7.22. Domains of the variables for the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint of Figure 3.7.22.
$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$
1$\left\{1,2\right\}$3$\left\{1,2,3\right\}$5$\left\{3,4,5,6\right\}$
2$\left\{1,2\right\}$4$\left\{2,3,4,5\right\}$
Table 3.7.22. Domains of the variables for the $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint of Figure 3.7.22. In addition the lower bound of the first argument of the $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint is equal to $\left\{{x}_{1},{x}_{2},{x}_{3}\right\}$.
$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$
1$\left\{1,2\right\}$3$\left\{1,2,3\right\}$5$\left\{3,4\right\}$
2$\left\{1,2\right\}$4$\left\{2,3\right\}$