CHIP

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

$\mathrm{𝚊𝚜𝚜𝚒𝚐𝚗𝚖𝚎𝚗𝚝}$, $\mathrm{𝚌𝚑𝚊𝚗𝚗𝚎𝚕}$, $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}\_\mathrm{𝚌𝚑𝚊𝚗𝚗𝚎𝚕𝚒𝚗𝚐}$.

The $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint holds since:

• $\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[1\right].\mathrm{𝚜𝚞𝚌𝚌}=2\iff \mathrm{𝙽𝙾𝙳𝙴𝚂}\left[2\right].\mathrm{𝚙𝚛𝚎𝚍}=1$,

• $\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[2\right].\mathrm{𝚜𝚞𝚌𝚌}=1\iff \mathrm{𝙽𝙾𝙳𝙴𝚂}\left[1\right].\mathrm{𝚙𝚛𝚎𝚍}=2$,

• $\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[3\right].\mathrm{𝚜𝚞𝚌𝚌}=5\iff \mathrm{𝙽𝙾𝙳𝙴𝚂}\left[5\right].\mathrm{𝚙𝚛𝚎𝚍}=3$,

• $\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[4\right].\mathrm{𝚜𝚞𝚌𝚌}=3\iff \mathrm{𝙽𝙾𝙳𝙴𝚂}\left[3\right].\mathrm{𝚙𝚛𝚎𝚍}=4$,

• $\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[5\right].\mathrm{𝚜𝚞𝚌𝚌}=4\iff \mathrm{𝙽𝙾𝙳𝙴𝚂}\left[4\right].\mathrm{𝚙𝚛𝚎𝚍}=5$.

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

• Attributes of $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ are permutable w.r.t. permutation $\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)$ $(\mathrm{𝚜𝚞𝚌𝚌},\mathrm{𝚙𝚛𝚎𝚍})$ (permutation applied to all items).

This constraint is used in order to make the link between the successor and the predecessor variables. This is sometimes required by specific heuristics that use both predecessor and successor variables. In some problems, the $\mathrm{successor}$ and $\mathrm{predecessor}$ variables are respectively interpreted as column an row variables (i.e., we have a bijection between the $\mathrm{successor}$ variables and their values). This is for instance the case in the $n$-queens problem (i.e., place $n$ queens on a $n$ by $n$ chessboard in such a way that no two queens are on the same row, the same column or the same diagonal) when we use the following model: to each column of the chessboard we associate a variable that gives the row where the corresponding queen is located. Symmetrically, to each row of the chessboard we create a variable that indicates the column where the associated queen is placed. Having these two sets of variables, we can now write a heuristics that selects the column or the row for which we have the fewest number of alternatives for placing a queen.

In the original $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint of CHIP the $\mathrm{𝚒𝚗𝚍𝚎𝚡}$ attribute was not explicitly present. It was implicitly defined as the position of a variable in a list, the first position being 1. This is also the case for SICStus Prolog, JaCoP and Gecode where the variables are respectively indexed from 1, 0 and 0. Within SICStus Prolog and JaCoP (http://www.jacop.eu/), the $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint is called $\mathrm{𝚊𝚜𝚜𝚒𝚐𝚗𝚖𝚎𝚗𝚝}$. Within Gecode, it is called $\mathrm{𝚌𝚑𝚊𝚗𝚗𝚎𝚕}$ (http://www.gecode.org/).

We can reuse the filtering algorithm associated with the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint, both for the $\mathrm{successor}$ and the $\mathrm{predecessor}$ variables. In addition, each time value $j$ is removed from the $i^{th}$ $\mathrm{successor}$ variable, we have to remove value $i$ from the $j^{th}$ $\mathrm{predecessor}$ variable. Similarly, each time value $i$ is removed from the $j^{th}$ $\mathrm{successor}$ variable, we have also to remove value $j$ from the $i^{th}$ $\mathrm{predecessor}$ variable.

inverseChanneling in Choco, channel in Gecode, assignment in SICStus.

common keyword: $\mathrm{𝚌𝚢𝚌𝚕𝚎}$, $\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (permutation).

generalisation: $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}\_\mathrm{𝚘𝚏𝚏𝚜𝚎𝚝}$ (do not assume anymore that the smallest value of the $\mathrm{𝚙𝚛𝚎𝚍}$ or $\mathrm{𝚜𝚞𝚌𝚌}$ attributes is equal to 1), $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}\_\mathrm{𝚜𝚎𝚝}$ ($\mathrm{𝚍𝚘𝚖𝚊𝚒𝚗}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ replaced by $\mathrm{𝚜𝚎𝚝}$ $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$), $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}\_\mathrm{𝚠𝚒𝚝𝚑𝚒𝚗}\_\mathrm{𝚛𝚊𝚗𝚐𝚎}$ (partial mapping between two collections of distinct size).

characteristic of a constraint: automaton, automaton with array of counters.

combinatorial object: permutation.

constraint type: graph constraint.

filtering: arc-consistency.

heuristics: heuristics.

modelling: channelling constraint, permutation channel, dual model.

modelling exercises: n-Amazon, zebra puzzle.

puzzles: n-Amazon, n-queen, zebra puzzle.

$\mathrm{𝙽𝙾𝙳𝙴𝚂}$

In order to express the binary constraint that links two vertices one has to make explicit the identifier of the vertices. This is why the $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint considers objects that have three attributes:

• One fixed attribute $\mathrm{𝚒𝚗𝚍𝚎𝚡}$ that is the identifier of the vertex,

• One variable attribute $\mathrm{𝚜𝚞𝚌𝚌}$ that is the successor of the vertex,

• One variable attribute $\mathrm{𝚙𝚛𝚎𝚍}$ that is the predecessor of the vertex.

Parts (A) and (B) of Figure 5.170.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{𝐍𝐀𝐑𝐂}$ graph property, the arcs of the final graph are stressed in bold.

Figure 5.170.1. Initial and final graph of the $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint

(a)	(b)

Since all the $\mathrm{𝚒𝚗𝚍𝚎𝚡}$ attributes of the $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ collection are distinct and because of the first condition $\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}$ of the arc constraint all the vertices of the final graph have at most one predecessor.

Since all the $\mathrm{𝚒𝚗𝚍𝚎𝚡}$ attributes of the $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ collection are distinct and because of the second condition $\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚙𝚛𝚎𝚍}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}$ of the arc constraint all the vertices of the final graph have at most one successor.

From the two previous remarks it follows that the final graph is made up from disjoint paths and disjoint circuits. Therefore the maximum number of arcs of the final graph is equal to its maximum number of vertices $\mathrm{𝙽𝙾𝙳𝙴𝚂}$. So we can rewrite the graph property $\mathrm{𝐍𝐀𝐑𝐂}=\left|\mathrm{𝙽𝙾𝙳𝙴𝚂}\right|$ to $\mathrm{𝐍𝐀𝐑𝐂}\ge \left|\mathrm{𝙽𝙾𝙳𝙴𝚂}\right|$ and simplify $\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}$ to $\overline{\mathrm{𝐍𝐀𝐑𝐂}}$.

Figure 5.170.2 depicts the automaton associated with the $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint. To each item of the collection $\mathrm{𝙽𝙾𝙳𝙴𝚂}$ corresponds a signature variable ${𝚂}_i$ that is equal to 1.

Figure 5.170.2. Automaton of the $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint