One-in-Two-Matching Problem is NP-complete
<|reference_start|>One-in-Two-Matching Problem is NP-complete: 2-dimensional Matching Problem, which requires to find a matching of left- to right-vertices 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 $i \leq n/2$ it is required that either $\pi(2i-1)=2i-1$ or $\pi(2i)=2i$, 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 One-in-Two Matching to 3-Dimensional Matching, we show that SAT can be linearly reduced to 3DM, while the original Karp reduction was only cubic.<|reference_end|>
@article{caracciolo2006one-in-two-matching,
    title={One-in-Two-Matching Problem is NP-complete},
    author={Sergio Caracciolo, Davide Fichera, Andrea Sportiello},
    journal={arXiv preprint arXiv:cs/0604113},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0604113},
    primaryClass={cs.CC}
}