One-in-Two-Matching Problem is NP-complete
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2006
- DOI:
- 10.48550/arXiv.cs/0604113
- arXiv:
- arXiv:cs/0604113
- Bibcode:
- 2006cs........4113C
- Keywords:
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- Computer Science - Computational Complexity;
- F.2.2
- E-Print:
- 30 pages