[PachetRoy99]

$\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂})$

$\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\_\mathrm{𝚘𝚗}\_\mathrm{𝚊𝚝𝚝𝚛𝚒𝚋𝚞𝚝𝚎𝚜}\_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$, $\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$.

The $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint holds since its first argument $\mathrm{𝙽𝚅𝙰𝙻}=4$ is set to the number of distinct values occurring within the collection $\langle 3,1,7,1,6\rangle$.

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

• All occurrences of two distinct values of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}$ can be swapped; all occurrences of a value of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}$ can be renamed to any unused value.

A classical example from the early 1850s is the dominating queens chess puzzle problem: Place a number of queens on a $n$ by $n$ chessboard in such a way that all squares are either attacked by a queen or are occupied by a queen. A queen can attack all squares located on the same column, on the same row or on the same diagonal. Part (A) of Figure 5.238.1 illustrates a set of five queens which together attack all of the squares of an 8 by 8 chessboard. The dominating queens problem can be modelled as one single $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint:

• We first label the different squares of the chessboard from 1 to $n^2$.

• We then associate to each square $S$ of the chessboard a domain variable. Its initial domain is set to the numbers of the squares that can be attacked from $S$. For instance, in the context of an 8 by 8 chessboard, the initial domain of $V_{29}$ will be set to {2,5,8,11,13,15,20..22,25..32,36..38,43,45,47,50,53,56,57,61} (see the green squares of part (B) of Figure 5.238.1).

• Finally, we post the constraint $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$$(Q,\langle \mathrm{𝚟𝚊𝚛}-V_1,\mathrm{𝚟𝚊𝚛}-V_2,...,\mathrm{𝚟𝚊𝚛}-V_{n^2}\rangle )$ where $Q$ is a domain variable in $[1,n^2]$ that gives the total number of queens used for controlling all squares of the chessboard. For the solution depicted by Part (A) of Figure 5.238.1, the number in each square of Part (C) of Figure 5.238.1 gives the value assigned to the corresponding variable. Note that, since a given square can be attacked by several queens, we have also other assignments corresponding to the solution depicted by Part (A) of Figure 5.238.1.

Figure 5.238.1. Modelling the dominating queens problem with one single $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint

The $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint occurs also in many practical applications. In the context of timetabling one wants to set up a limit on the maximum number of activity types it is possible to perform. For frequency allocation problems, one optimisation criteria corresponds to the fact that you want to minimise the number of distinct frequencies that you use all over the entire network. The $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint generalises several constraints like:

• $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)$: in order to get the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint, one has to set $\mathrm{𝙽𝚅𝙰𝙻}$ to the total number of variables.

• $\mathrm{𝚗𝚘𝚝}\_\mathrm{𝚊𝚕𝚕}\_\mathrm{𝚎𝚚𝚞𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)$: in order to get the $\mathrm{𝚗𝚘𝚝}\_\mathrm{𝚊𝚕𝚕}\_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ constraint, one has to set the minimum value of $\mathrm{𝙽𝚅𝙰𝙻}$ to 2.

This constraint appears in [PachetRoy99] under the name of Cardinality on Attributes Values. The $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint is called $\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$ in JaCoP (http://www.jacop.eu/). A constraint called $𝚔\_\mathrm{𝚍𝚒𝚏𝚏}$ enforcing that a set of variables takes at least $k$ distinct values appears in the PhD thesis of J.-C. Régin [Regin95].

It was shown in [BessiereHebrardHnichWalshO4] that, finding out whether a $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint has a solution or not is NP-hard. This was achieved by reduction from 3-SAT. In the same article, it is also shown, by reduction from minimum hitting set cardinality, that computing a sharp lower bound on $\mathrm{𝙽𝚅𝙰𝙻}$ is NP-hard.

Both reformulations of the $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint and of the $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$ constraint use the $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint.

A first filtering algorithm for the $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint was described in [Beldiceanu01]. Assuming that the minimum value of variable $\mathrm{𝙽𝚅𝙰𝙻}$ is not constrained at all, two algorithms that both achieve bound-consistency were provided one year later in [BeldiceanuCarlssonThiel02]. Under the same assumption, algorithms that partially take into account holes in the domains of the variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection are described in [BeldiceanuCarlssonThiel02], [BessiereHebrardHnichKiziltanWalsh05].

A model, involving linear inequalities constraints, preserving bound-consistency was introduced in [BessiereKatsirelosNarodytskaQuimperWalsh10CP].

nvalue in SICStus.

$\mathrm{𝚝𝚛𝚊𝚌𝚔}$.

assignment dimension added: $\mathrm{𝚊𝚜𝚜𝚒𝚐𝚗}\_\mathrm{𝚊𝚗𝚍}\_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}$.

common keyword: $\mathrm{𝚊𝚖𝚘𝚗𝚐}$, $\mathrm{𝚊𝚖𝚘𝚗𝚐}\_\mathrm{𝚍𝚒𝚏𝚏}\_\mathtt{0}$, $\mathrm{𝚌𝚘𝚞𝚗𝚝}$, $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$, $\mathrm{𝚖𝚊𝚡}\_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$, $\mathrm{𝚖𝚒𝚗}\_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ (counting constraint), $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}\_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}\_\mathtt{0}$ (counting constraint,number of distinct values).

cost variant: $\mathrm{𝚜𝚞𝚖}\_\mathrm{𝚘𝚏}\_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚜}\_\mathrm{𝚘𝚏}\_\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}\_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$ (introduce a weight for each value and replace number of distinct values by sum of weights associated with distinct values).

generalisation: $\mathrm{𝚗𝚌𝚕𝚊𝚜𝚜}$ ($\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ replaced by $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}\in \mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}$), $\mathrm{𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎}$ ($\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ replaced by $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}\mathrm{mod}\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}$), $\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ ($\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ replaced by $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}/\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}$), $\mathrm{𝚗𝚙𝚊𝚒𝚛}$ ($\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$ replaced by $\mathrm{𝚙𝚊𝚒𝚛}$ of $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}$), $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}$ (replace an equality with the number of distinct values by a comparison with the number of distinct values), $\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}$ (variable replaced by vector).

implied by: $\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}\_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$.

implies: $\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}\_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ ($=\mathrm{𝙽𝚅𝙰𝙻}$ replaced by $\ge \mathrm{𝙽𝚅𝙰𝙻}$), $\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ ($=\mathrm{𝙽𝚅𝙰𝙻}$ replaced by $\le \mathrm{𝙽𝚅𝙰𝙻}$).

related: $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ (restrict number of distinct colours on each maximum clique of the interval graph associated with the tasks), $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$ (restrict number of distinct colours on each maximum clique of the interval graph associated with the tasks assigned to the same machine), $\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}\_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\_\mathrm{𝚌𝚑𝚊𝚒𝚗}$, $𝚔\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (necessary condition for two overlapping $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraints), $\mathrm{𝚜𝚘𝚏𝚝}\_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}\_\mathrm{𝚟𝚊𝚛}$.

shift of concept: $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\_\mathrm{𝚘𝚗}\_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚜𝚎𝚌𝚝𝚒𝚘𝚗}$.

soft variant: $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}\_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}\_\mathtt{0}$ (value 0 is ignored).

specialisation: $\mathrm{𝚊𝚕𝚕}\_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ (enforce to have one single value), $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (enforce a number of distinct values equal to the number of variables), $\mathrm{𝚗𝚘𝚝}\_\mathrm{𝚊𝚕𝚕}\_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ (enforce to have at least two distinct values).

uses in its reformulation: $\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}\_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$, $\mathrm{𝚖𝚒𝚗}\_𝚗$.

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

complexity: 3-SAT, minimum hitting set cardinality.

constraint type: counting constraint, value partitioning constraint.

filtering: bound-consistency, convex bipartite graph.

final graph structure: strongly connected component, equivalence.

modelling: number of distinct equivalence classes, number of distinct values.

problems: domination.

puzzles: dominating queens.

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$

Parts (A) and (B) of Figure 5.238.2 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{𝐍𝐒𝐂𝐂}$ graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a value that is assigned to some variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection. The 4 following values 1, 3, 6 and 7 are used by the variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection.

Figure 5.238.2. Initial and final graph of the $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint

(a)	(b)

Figure 5.238.3 depicts the automaton associated with the $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint. To each item of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ corresponds a signature variable ${𝚂}_i$ that is equal to 0.

Figure 5.238.3. Automaton of the $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ constraint