Name of an Exact Cover by 3 sets variant

Question

Exact cover by 3-sets is $\sf{NP}$-complete:

Instance: Given a finite set $X = \{ x_1,x_2,...,x_{3n}\}$ of $3n$ elements and a collection $C = \{ ( x_{i_1}, x_{i_2}, x_{i_3}) \} $ of $m$ 3-elements subsets of $X$;
Question: Find a subcollection $C'$ of $C$ such that every element in $X$ is contained in exactly one member of $C'$.

The problem remains NPC even if we add the following condition:

every element of $X$ appears exactly in three subsets of $C$

Has this variant an "official" name?

It's similar to $3$-uniform hypergraph maximum matching. Perhaps in your case it is actually true that $i_1 \in {1,\ldots,n}$, $i_2 \in {n+1,\ldots,2n}$ and $i_3 \in {2n+1,\ldots,3n}$? — Yuval Filmus, Jun 10 '13 at 18:12
@YuvalFilmus: it is slightly more general: $i_j \in {1,2,...,3n}$. — Vor, Jun 10 '13 at 18:53

score 3 · Accepted Answer · answered Jun 11 '13 at 20:52

3

Gonzalez called this variant RXC3, for "Restricted Exact Cover by Three Sets."

answered Jun 11 '13 at 20:52

rphv

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thank you, I already saw it ... however if I don't receive other answers in 1-2 days I'll accept yours! – Vor Jun 12 '13 at 14:12

Name of an Exact Cover by 3 sets variant

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