Abstract
IV-matching is a generalization of perfect bipartite matching. The complexity of finding IV-matching in a graph was posted as an open problem in [Fiala, Klavik, Kratochvil, Nedela, ICALP 2014].
It was shown that if it is possible to find an IV-matching in a polynomial time, then the described algorithm can be modified to run in polynomial time. In this note, we resolve the question and prove that, contrary to the expectations of the authors, the given problem is strongly NP-hard (already in the simplest non-trivial case of three layers).
Hence it is unlikely that there would be an efficient (polynomial or pseudo polynomial in case of at least four layers) algorithm solving the problem.