The numerical three-dimensional matching problem is a decision problem in which three disjoint sets 
, 
, and 
each contain 
elements, every element 
has a positive integer
weight 
, and 
is a positive integer.
The question is whether 
has a set partition
into 
triples, each containing one element from each of 
, 
, and 
, such that every triple has total weight 
(Garey and Johnson 1979, p. 224).
Equivalently, the permitted triples form the set
 |
and the question is whether 
contains a solution to the three-dimensional
matching problem. The numerical problem is NP-complete
even when the weights and 
are bounded by a polynomial
in 
(Garey and Johnson 1979). Elements remain distinct even when
their weights coincide.
For example, let the weights in 
be 1 and 2, those in 
be 3 and 4, and those in 
be 6 and 4. For 
, a solution consists of triples with weights 
and 
.
