Numerical 3-dimensional matching

Numerical 3-dimensional matching is a strongly 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 ^[1].

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.

Proof of NP-completeness

NP-completeness of the 3-partition problem is stated by Garey and Johnson in "Computers and Intractability; A Guide to the Theory of NP-Completeness".^[1] It is done by a reduction from 3-dimensional matching via 4-partition. To prove NP-completeness of the numerical 3-dimensional matching, the proof is similar, but a reduction from 3-dimensional matching via the numerical 4-dimensional matching problem should be used.

References

1. ^ ^{a b} Garey, Michael R. and David S. Johnson (1979), Computers and Intractability; A Guide to the Theory of NP-Completeness. ISBN 0-7167-1045-5