Numerical 3-dimensional matching

Numerical 3-dimensional matching is an NP-complete decision problem. It is given by three multisets of integers ${\displaystyle X}$, ${\displaystyle Y}$ and ${\displaystyle Z}$, each containing ${\displaystyle k}$ elements, and a bound ${\displaystyle b}$. The goal is to select a subset ${\displaystyle M}$ of ${\displaystyle X\times Y\times Z}$ such that every integer in ${\displaystyle X}$, ${\displaystyle Y}$ and ${\displaystyle Z}$ occurs exactly once and that for every triple ${\displaystyle (x,y,z)}$ in the subset ${\displaystyle x+y+z=b}$ holds. This problem is labeled as [SP16] in.^[1]

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

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 ${\displaystyle X}$, ${\displaystyle Y}$ and ${\displaystyle Z}$, where there is an hyperedge ⁠${\displaystyle (x,y,z)}$⁠ if and only if ${\displaystyle x+y+z=T}$. A matching in this hypergraph corresponds to a solution to ABC-partition.

The numerical 3-d matching problem is problem [SP16] of Garey and Johnson.^[1] They claim it is NP-complete, and refer to,^[2] but the claim is not proved at that source. The NP-hardness of the related problem 3-partition is done in ^[1] 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^[3] 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^[4] 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.