It is NP-hard to approximate maximum 3D matching problem even if each element occurs exactly in two triples. I'm interested in the following decision version of 3D matching.
Informally, Given a set of triples $F$ of elements such that each element occurs exactly in two triples, Is there a subset of $F$ such that each element occurs in exactly one triple?
Is this decision problem solvable in polynomial time? Is it $NP$-complete?