Global Constraint Catalog: Cgroup_skip_isolated

5.173. group_skip_isolated_item

DESCRIPTION

LINKS

GRAPH

AUTOMATON

Origin

Constraint

$𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖 (\begin{matrix} 𝙽𝙶𝚁𝙾𝚄𝙿, \\ 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴, \\ 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴, \\ 𝙽𝚅𝙰𝙻, \\ 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, \\ 𝚅𝙰𝙻𝚄𝙴𝚂 \end{matrix})$

Arguments

$𝙽𝙶𝚁𝙾𝚄𝙿$	$𝚍𝚟𝚊𝚛$
$𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$	$𝚍𝚟𝚊𝚛$
$𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$	$𝚍𝚟𝚊𝚛$
$𝙽𝚅𝙰𝙻$	$𝚍𝚟𝚊𝚛$
$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$	$𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛 - 𝚍𝚟𝚊𝚛)$
$𝚅𝙰𝙻𝚄𝙴𝚂$	$𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚕 - 𝚒𝚗𝚝)$

Restrictions

𝙽𝙶𝚁𝙾𝚄𝙿 \geq 0

3 * 𝙽𝙶𝚁𝙾𝚄𝙿 \leq | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | + 1

𝙼𝙸𝙽_𝚂𝙸𝚉𝙴 \geq 0

𝙼𝙸𝙽_𝚂𝙸𝚉𝙴 \neq 1

𝙼𝙰𝚇_𝚂𝙸𝚉𝙴 \geq 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴

𝙽𝚅𝙰𝙻 \geq 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴

𝙽𝚅𝙰𝙻 \geq 𝙽𝙶𝚁𝙾𝚄𝙿

𝙽𝚅𝙰𝙻 \leq | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚟𝚊𝚛)

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝙻𝚄𝙴𝚂, 𝚟𝚊𝚕)

𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝

(𝚅𝙰𝙻𝚄𝙴𝚂, 𝚟𝚊𝚕)

Purpose

Let $n$ be the number of variables of the collection $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ . Let $X_{i}, X_{i + 1}, \dots, X_{j}$ $(1 \leq i < j \leq n)$ be consecutive variables of the collection of variables $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ such that the following conditions apply:

All variables $X_{i}, \dots, X_{j}$ take their value in the set of values $𝚅𝙰𝙻𝚄𝙴𝚂$ ,
$i = 1$ or $X_{i - 1}$ does not take a value in $𝚅𝙰𝙻𝚄𝙴𝚂$ ,
$j = n$ or $X_{j + 1}$ does not take a value in $𝚅𝙰𝙻𝚄𝙴𝚂$ .

We call such a set of variables a group. The constraint $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ is true if all the following conditions hold:

There are exactly $𝙽𝙶𝚁𝙾𝚄𝙿$ groups of variables,
The number of variables of the smallest group is $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ ,
The number of variables of the largest group is $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ ,
The number of variables that take their value in the set of values $𝚅𝙰𝙻𝚄𝙴𝚂$ is equal to $𝙽𝚅𝙰𝙻$ .

Example

(1, 2, 2, 3, 〈2, 8, 1, 7, 4, 5, 1, 1, 1〉, 〈0, 2, 4, 6, 8〉)

Given the fact that groups are formed by even values in ${0, 2, 4, 6, 8}$ (i.e., values expressed by the $𝚅𝙰𝙻𝚄𝙴𝚂$ collection), and the fact that isolated even values are ignored, the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint holds since:

Its first argument, $𝙽𝙶𝚁𝙾𝚄𝙿$ , is set to value 1 since the sequence $281745111$ contains only one group of even values involving more than one even value (i.e., group $28$ ).
Its second and third arguments, $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ and $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ , are both set to 2 since the only group of even values with more than one even value involves two values (i.e., group $28$ ).
The fourth argument, $𝙽𝚅𝙰𝙻$ , is fixed to 2 since it corresponds to the total number of even values belonging to groups involving more than one even value (i.e., value 4 is discarded since it is an isolated even value of the sequence $281745111$ ).

Typical

𝙽𝙶𝚁𝙾𝚄𝙿 > 0

𝙼𝙸𝙽_𝚂𝙸𝚉𝙴 > 0

𝙽𝚅𝙰𝙻 > 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴

𝙽𝚅𝙰𝙻 > 𝙽𝙶𝚁𝙾𝚄𝙿

𝙽𝚅𝙰𝙻 < | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 1

𝚛𝚊𝚗𝚐𝚎

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛) > 1

| 𝚅𝙰𝙻𝚄𝙴𝚂 | > 0

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > | 𝚅𝙰𝙻𝚄𝙴𝚂 |

Symmetries

Items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ can be reversed.
Items of $𝚅𝙰𝙻𝚄𝙴𝚂$ are permutable.
An occurrence of a value of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛$ that belongs to $𝚅𝙰𝙻𝚄𝙴𝚂 . 𝚟𝚊𝚕$ (resp. does not belong to $𝚅𝙰𝙻𝚄𝙴𝚂 . 𝚟𝚊𝚕$ ) can be replaced by any other value in $𝚅𝙰𝙻𝚄𝙴𝚂 . 𝚟𝚊𝚕$ (resp. not in $𝚅𝙰𝙻𝚄𝙴𝚂 . 𝚟𝚊𝚕$ ).

Arg. properties

Functional dependency: $𝙽𝙶𝚁𝙾𝚄𝙿$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝚅𝙰𝙻𝚄𝙴𝚂$ .
Functional dependency: $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝚅𝙰𝙻𝚄𝙴𝚂$ .
Functional dependency: $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝚅𝙰𝙻𝚄𝙴𝚂$ .
Functional dependency: $𝙽𝚅𝙰𝙻$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝚅𝙰𝙻𝚄𝙴𝚂$ .

Usage

This constraint is useful in order to specify rules about how rest days should be allocated to a person during a period of $n$ consecutive days. In this case $𝚅𝙰𝙻𝚄𝙴𝚂$ are the codes for the rest days (perhaps a single value) and $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ corresponds to the amount of work done during $n$ consecutive days. We can then express a rule like: in a month one should have at least 4 periods of at least 2 rest days (isolated rest days are not counted as rest periods).

Remark

The following invariant imposes a limit on the maximum number of groups wrt the minimum size of a group and the total number of variables: $𝙽𝙶𝚁𝙾𝚄𝙿 \cdot (max (𝙼𝙸𝙽_𝚂𝙸𝚉𝙴, 2) + 1) \leq | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | + 1$ .

used in graph description: $𝚒𝚗$ .

Keywords

characteristic of a constraint: automaton, automaton with counters, automaton with same input symbol.

combinatorial object: sequence.

constraint arguments: reverse of a constraint.

constraint network structure: alpha-acyclic constraint network(2), alpha-acyclic constraint network(3).

constraint type: timetabling constraint.

filtering: glue matrix.

final graph structure: strongly connected component.

modelling: functional dependency.

Arc input(s)

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$

Arc generator

𝐶𝐻𝐴𝐼𝑁

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1, 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2)

Arc arity

Arc constraint(s)

•

𝚒𝚗

(𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚟𝚊𝚛, 𝚅𝙰𝙻𝚄𝙴𝚂)

•

𝚒𝚗

(𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚟𝚊𝚛, 𝚅𝙰𝙻𝚄𝙴𝚂)

Graph property(ies)

•

𝐍𝐒𝐂𝐂

= 𝙽𝙶𝚁𝙾𝚄𝙿

•

𝐌𝐈𝐍_𝐍𝐒𝐂𝐂

= 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴

•

𝐌𝐀𝐗_𝐍𝐒𝐂𝐂

= 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴

•

𝐍𝐕𝐄𝐑𝐓𝐄𝐗

= 𝙽𝚅𝙰𝙻

Graph model

We use the $𝐶𝐻𝐴𝐼𝑁$ arc generator in order to produce the initial graph. In the context of the Example slot, this creates the graph depicted in part (A) of Figure 5.173.1. We use $𝐶𝐻𝐴𝐼𝑁$ together with the arc constraint $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚟𝚊𝚛 \in 𝚅𝙰𝙻𝚄𝙴𝚂 \land 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚟𝚊𝚛 \in 𝚅𝙰𝙻𝚄𝙴𝚂$ in order to skip the isolated variables that take a value in $𝚅𝙰𝙻𝚄𝙴𝚂$ that we do not want to count as a group. This is why, on the example, value 4 is not counted as a group. Part (B) of Figure 5.173.1 shows the final graph associated with the Example slot. The $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint of the Example slot holds since:

The final graph contains one strongly connected component. Therefore the number of groups is equal to one.
The unique strongly connected component of the final graph contains two vertices. Therefore $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ and $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ are both equal to 2.
The number of vertices of the final graph is equal to two. Therefore $𝙽𝚅𝙰𝙻$ is equal to 2.

Figure 5.173.1. Initial and final graph of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint


(a)	(b)

Automaton: Figures 5.173.2, 5.173.4, 5.173.6 and 5.173.8 depict the different automata associated with the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint. For the automata that respectively compute $𝙽𝙶𝚁𝙾𝚄𝙿$ , $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ , $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ and $𝙽𝚅𝙰𝙻$ we have a 0-1 signature variable $𝚂_{i}$ for each variable ${𝚅𝙰𝚁}_{i}$ of the collection $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ . The following signature constraint links ${𝚅𝙰𝚁}_{i}$ and $𝚂_{i}$ : ${𝚅𝙰𝚁}_{i} \in 𝚅𝙰𝙻𝚄𝙴𝚂 \Leftrightarrow 𝚂_{i}$ .

Figure 5.173.2. Automaton for the $𝙽𝙶𝚁𝙾𝚄𝙿$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.3. Hypergraph of the reformulation corresponding to the automaton (with one counter) of the $𝙽𝙶𝚁𝙾𝚄𝙿$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint (since all states of the automaton are accepting there is no restriction on the last variable $Q_{n}$ )

Figure 5.173.4. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.5. Hypergraph of the reformulation corresponding to the automaton (with two counters) of the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint where $𝙽$ stands for $| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |$ (since all states of the automaton are accepting there is no restriction on the last variable $Q_{n}$ )

Figure 5.173.6. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.7. Hypergraph of the reformulation corresponding to the automaton (with two counters) of the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint (since all states of the automaton are accepting there is no restriction on the last variable $Q_{n}$ )

Figure 5.173.8. Automaton for the $𝙽𝚅𝙰𝙻$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.9. Hypergraph of the reformulation corresponding to the automaton (with one counter) of the $𝙽𝚅𝙰𝙻$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint

Global Constraint Catalog

5. Global Constraint Catalogue

5.173. group_skip_isolated_item

Figure 5.173.1. Initial and final graph of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint

Figure 5.173.2. Automaton for the $𝙽𝙶𝚁𝙾𝚄𝙿$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.4. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.6. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.8. Automaton for the $𝙽𝚅𝙰𝙻$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.9. Hypergraph of the reformulation corresponding to the automaton (with one counter) of the $𝙽𝚅𝙰𝙻$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint

Global Constraint Catalog

5. Global Constraint Catalogue

5.173. group_skip_isolated_item

Figure 5.173.1. Initial and final graph of the 𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖 constraint

Figure 5.173.2. Automaton for the 𝙽𝙶𝚁𝙾𝚄𝙿 argument of the 𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖 constraint and its glue matrix

Figure 5.173.4. Automaton for the 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴 argument of the 𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖 constraint and its glue matrix

Figure 5.173.6. Automaton for the 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴 argument of the 𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖 constraint and its glue matrix

Figure 5.173.8. Automaton for the 𝙽𝚅𝙰𝙻 argument of the 𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖 constraint and its glue matrix

Figure 5.173.9. Hypergraph of the reformulation corresponding to the automaton (with one counter) of the 𝙽𝚅𝙰𝙻 argument of the 𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖 constraint

Figure 5.173.1. Initial and final graph of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint

Figure 5.173.2. Automaton for the $𝙽𝙶𝚁𝙾𝚄𝙿$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.4. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.6. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.8. Automaton for the $𝙽𝚅𝙰𝙻$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint and its glue matrix

Figure 5.173.9. Hypergraph of the reformulation corresponding to the automaton (with one counter) of the $𝙽𝚅𝙰𝙻$ argument of the $𝚐𝚛𝚘𝚞𝚙_𝚜𝚔𝚒𝚙_𝚒𝚜𝚘𝚕𝚊𝚝𝚎𝚍_𝚒𝚝𝚎𝚖$ constraint