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 $π(2i-1)=2i-1$ 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 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.
A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter
在图中，匹配割是匹配的边割。匹配割是判定给定图是否有匹配割的问题，即使仅限于二分图，也是NP完全的。证明了直径为2的图的匹配割是多项式可解的。本文证明了对任意固定的整数$d\ge 3$，匹配割在直径为$d$的图类中是NP-完全的.这解决了Borowiecki和Jesse-Józefczyk在[Matching cutsets in graphs of diameter $2$，Theoretical计算机科学407（2008）574-582]中提出的公开问题。然后，我们表明，对于任何固定的整数$d\ge 4$，匹配割是NP-完全的，即使限制在类的二部图的直径$d$。作为对硬度结果的补充，我们证明了匹配割在直径不超过3的二部图类中是多项式时间可解的，并提出了一种新的求解直径为$2$的图类中匹配割的简单多项式时间算法.
