## 5.423. zero_or_not_zero_vectors

Origin

Tournament scheduling

Constraint

$\mathrm{𝚣𝚎𝚛𝚘}_\mathrm{𝚘𝚛}_\mathrm{𝚗𝚘𝚝}_\mathrm{𝚣𝚎𝚛𝚘}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}\right)$

Synonyms

$\mathrm{𝚣𝚎𝚛𝚘𝚜}_\mathrm{𝚘𝚛}_\mathrm{𝚗𝚘𝚝}_\mathrm{𝚣𝚎𝚛𝚘𝚜}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$, $\mathrm{𝚗𝚘𝚝}_\mathrm{𝚣𝚎𝚛𝚘}_\mathrm{𝚘𝚛}_\mathrm{𝚣𝚎𝚛𝚘}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$, $\mathrm{𝚗𝚘𝚝}_\mathrm{𝚣𝚎𝚛𝚘𝚜}_\mathrm{𝚘𝚛}_\mathrm{𝚣𝚎𝚛𝚘𝚜}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$.

Type
 $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Argument
 $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚎𝚌}-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\right)$
Restrictions
 $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|\ge 1$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚟𝚊𝚛}\right)$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|\ge 1$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$ $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$
Purpose

Given a collection of vectors enforces for each vector that either all its components are equal to 0, or all its components are different from 0. In addition imposes that at least one 0 is used.

Example
$\left(\begin{array}{c}〈\begin{array}{c}\mathrm{𝚟𝚎𝚌}-〈5,6〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈5,6〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈0,0〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈9,3〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈0,0〉\hfill \end{array}〉\hfill \end{array}\right)$

The $\mathrm{𝚣𝚎𝚛𝚘}_\mathrm{𝚘𝚛}_\mathrm{𝚗𝚘𝚝}_\mathrm{𝚣𝚎𝚛𝚘}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$ constraint holds since:

• Both components of the first vector $〈5,6〉$ are different from 0.

• Both components of the second vector $〈5,6〉$ are different from 0.

• Both components of the third vector $〈0,0〉$ are equal to 0.

• Both components of the fourth vector $〈9,3〉$ are different from 0.

• Both components of the fifth vector $〈0,0〉$ are equal to 0.

Typical
 $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|>1$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|>1$
Arg. properties

Contractible wrt. $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$.

Keywords