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).

• Functional dependency: $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}$ determined by $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}$ and $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚙𝚛𝚎𝚍}$.

• Functional dependency: $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚙𝚛𝚎𝚍}$ determined by $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}$ and $\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}$.

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 an $n$ 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/).

An arc-consistency filtering algorithm for the $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint is described in [Cymer12], [CymerPhD13]. The algorithm is based on the following ideas:

• We first normalize the domains of the variables by removing value $i$ from the $j^{th}$ $\mathrm{predecessor}$ variable if value $j$ does not belong to the $i^{th}$ $\mathrm{successor}$ variable, and by removing value $j$ from the $i^{th}$ $\mathrm{successor}$ variable if value $i$ does not belong to the $j^{th}$ $\mathrm{predecessor}$ variable.

• Second, one can map solutions to the $\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}$ constraint to perfect matchings in a so-called variable bipartite graph derived from the domain of the variables of the constraint in the following way: to each successor variable corresponds a vertex; similarly to each predecessor variable corresponds a vertex; there is and edge between the $i^{th}$ $\mathrm{successor}$ variable and the $j^{th}$ $\mathrm{predecessor}$ variable if and only if value $i$ belongs to the domain of the $j^{th}$ $\mathrm{predecessor}$ variable and value $j$ belongs to the domain of the $i^{th}$ $\mathrm{successor}$ variable.

• Third, Dulmage-Mendelsohn decomposition [DulmageMendelsohn58] is used to characterise all edges that do not belong to any perfect matching, and therefore prune the corresponding variables.

inverseChanneling in Choco, channel in Gecode, inverse in MiniZinc, 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).

implies (items to collection): $\mathrm{𝚕𝚎𝚡}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$.

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

combinatorial object: permutation.

constraint arguments: pure functional dependency.

constraint type: graph constraint.

filtering: bipartite matching, arc-consistency.

heuristics: heuristics.

modelling: channelling constraint, permutation channel, dual model, functional dependency.

modelling exercises: n-Amazons, zebra puzzle.

puzzles: n-Amazons, n-queens, zebra puzzle.