Numerical 3-dimensional matching

Numerical 3-dimensional matching is an NP-complete decision problem. It is given by three multisets of integers $$X$$, $$Y$$ and $$Z$$, each containing $$k$$ elements, and a bound $$b$$. The goal is to select a subset $$M$$ of $$X\times Y\times Z$$ such that every integer in $$X$$, $$Y$$ and $$Z$$ occurs exactly once and that for every triple $$(x,y,z)$$ in the subset $$x+y+z=b$$ holds. This problem is labeled as [SP16] in.

Example
Take $$X=\{3,4,4\}$$, $$Y=\{1,4,6\}$$ and $$Z=\{1,2,5\}$$, and $$b=10$$. This instance has a solution, namely $$\{(3,6,1), (4,4,2), (4,1,5)\}$$. Note that each triple sums to $$b=10$$. The set $$\{(3,6,1), (3,4,2), (4,1,5)\}$$ is not a solution for several reasons: not every number is used (a $$4\in X$$ is missing), a number is used too often (the $$3\in X$$) and not every triple sums to $$b$$ (since $$3+4+2=9\neq b=10$$). However, there is at least one solution to this problem, which is the property we are interested in with decision problems. If we would take $$b=11$$ for the same $$X$$, $$Y$$ and $$Z$$, this problem would have no solution (all numbers sum to $$30$$, which is not equal to $$k\cdot b=33$$ in this case).

Related problems
Every instance of the Numerical 3-dimensional matching problem is an instance of both the 3-partition problem, and the 3-dimensional matching problem.

Given an instance of numeric 3d-matching, construct a tripartite hypergraph with sides $$X$$, $$Y$$ and $$Z$$, where there is an hyperedge $(x, y, z)$ if and only if $$x+y+z = T$$. A matching in this hypergraph corresponds to a solution to ABC-partition.

Proof of NP-completeness
The numerical 3-d matching problem is problem [SP16] of Garey and Johnson. They claim it is NP-complete, and refer to, but the claim is not proved at that source. The NP-hardness of the related problem 3-partition is done in by a reduction from 3-dimensional matching via 4-partition. To prove NP-completeness of the numerical 3-dimensional matching, the proof should be similar, but a reduction from 3-dimensional matching via the numerical 4-dimensional matching problem should be used. Explicit proofs of NP-hardness are given in later papers:


 * Yu, Hoogeveen and Lenstra prove NP-hardness of a very restricted version of Numerical 3-Dimensional Matching, in which two of the three sets contain only the numbers 1,...,k.
 * Caracciolo, Fichera, and Sportiello prove NP-hardness of Numerical 3-Dimensional Matching and related problems by reduction from NAE-SAT. The reduction is linear, that is, the size of the reduced instance is a linear function of the size of the original instance.