One-in-Two-Matching Problem is NP-complete

2-dimensional Matching Problem, which requires to find a matching of leftto rightvertices in a balanced 2n-vertex bipartite graph, is a well-known polynomial problem, while various variants, like the 3-dimensional analogoue (3DM, with triangles on a tripartite graph), or the Hamiltonian Circuit Problem (HC, a restriction to “unicyclic” matchings) are among the main examples of NP-hard problems, since the first Karp reduction series of 1972. The same holds for the weighted variants of these problems, the Linear Assignment Problem being polynomial, and the Numerical 3-Dimensional Matching and Travelling Salesman Problem being NP-complete. In this paper we show that a small modification of the 2-dimensional Matching and Assignment Problems in which for each i ≤ n/2 it is required that either π(2i− 1) = 2i− 1 or π(2i) = 2i, is a NP-complete problem. The proof is by linear reduction from SAT (or NAE-SAT), with the size n of the Matching Problem being four times the number of edges in the factor graph representation of the boolean problem. As a corollary, in combination with the simple linear reduction of Onein-Two Matching to 3-Dimensional Matching, we show that SAT can be linearly reduced to 3DM, while the original Karp reduction was only cubic.