Global Constraint Catalog: Csum_squares

5.393. sum_squares_ctr

DESCRIPTION

LINKS

Origin

Arithmetic constraint.

Constraint

$𝚜𝚞𝚖_𝚜𝚚𝚞𝚊𝚛𝚎𝚜_𝚌𝚝𝚛 (𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝙲𝚃𝚁, 𝚅𝙰𝚁)$

Synonyms

$𝚜𝚞𝚖_𝚜𝚚𝚞𝚊𝚛𝚎𝚜$ , $𝚜𝚞𝚖_𝚘𝚏_𝚜𝚚𝚞𝚊𝚛𝚎𝚜$ , $𝚜𝚞𝚖_𝚘𝚏_𝚜𝚚𝚞𝚊𝚛𝚎𝚜_𝚌𝚝𝚛$ .

Arguments

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$	$𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛 - 𝚍𝚟𝚊𝚛)$
$𝙲𝚃𝚁$	$𝚊𝚝𝚘𝚖$
$𝚅𝙰𝚁$	$𝚍𝚟𝚊𝚛$

Restrictions

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚟𝚊𝚛)

𝙲𝚃𝚁 \in [=, \neq, <, \geq, >, \leq]

Purpose

Constraint the sum of the squares of a set of domain variables. More precisely, let $𝚂$ denote the sum of the squares of the variables of the $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ collection (when the collection is empty the corresponding sum is equal to 0). Enforce the following constraint to hold: $𝚂 𝙲𝚃𝚁 𝚅𝙰𝚁$ .

Example

(〈1, 1, 4〉, =, 18)

The $𝚜𝚞𝚖_𝚜𝚚𝚞𝚊𝚛𝚎𝚜_𝚌𝚝𝚛$ constraint holds since the condition $1^{2} + 1^{2} + 4^{2} = 18$ is satisfied.

Typical

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 1

𝚛𝚊𝚗𝚐𝚎

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛) > 1

𝙲𝚃𝚁 \in [=, <, \geq, >, \leq]

Symmetry

Items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ are permutable.

Arg. properties

Contractible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝙲𝚃𝚁 \in [<, \leq]$ .
Extensible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝙲𝚃𝚁 \in [\geq, >]$ .
Aggregate: $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 (𝚞𝚗𝚒𝚘𝚗)$ , $𝙲𝚃𝚁 (𝚒𝚍)$ , $𝚅𝙰𝚁 (+)$ .

Keywords

characteristic of a constraint: sum.

constraint type: predefined constraint, arithmetic constraint.

Cond. implications

$•$ $𝚜𝚞𝚖_𝚜𝚚𝚞𝚊𝚛𝚎𝚜_𝚌𝚝𝚛 (𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝙲𝚃𝚁, 𝚅𝙰𝚁)$

with $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \geq - 1$

and $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \leq 1$

implies $𝚜𝚞𝚖_𝚙𝚘𝚠𝚎𝚛𝚜 4_𝚌𝚝𝚛$ $(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝙲𝚃𝚁, 𝚅𝙰𝚁)$

when $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \geq - 1$

and $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \leq 1$ .

$•$ $𝚜𝚞𝚖_𝚜𝚚𝚞𝚊𝚛𝚎𝚜_𝚌𝚝𝚛 (𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝙲𝚃𝚁, 𝚅𝙰𝚁)$

with $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \geq - 1$

and $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \leq 1$

implies $𝚜𝚞𝚖_𝚙𝚘𝚠𝚎𝚛𝚜 6_𝚌𝚝𝚛$ $(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝙲𝚃𝚁, 𝚅𝙰𝚁)$

when $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \geq - 1$

and $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛 \leq 1$ .

Global Constraint Catalog

5. Global Constraint Catalogue

5.393. sum_squares_ctr