IV-matching is strongly NP-hard

TL;DR

This paper proves that IV-matching, a generalization of perfect bipartite matching, is strongly NP-hard even in the simplest non-trivial case, resolving an open problem from ICALP 2014.

Contribution

It establishes the strong NP-hardness of IV-matching, a previously unresolved complexity question, even for four-layer graphs.

Findings

IV-matching is strongly NP-hard in the simplest case of four layers

The problem is unlikely to have an efficient polynomial or pseudo-polynomial algorithm

Resolves an open problem from ICALP 2014 about the complexity of IV-matching

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 at the ICALP 2014 conference. 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 four layers). Hence it is unlikely that there would be an efficient (polynomial or pseudo-polynomial) algorithm solving the problem.