New algorithms and complexity status of the reducibility problem of sequences in open shop scheduling minimizing the makespan

We consider the problem of minimizing the makespan in open shop scheduling. The decision problem whether a given sequence in open shop scheduling is irreducible has already been considered, however, has not been solved yet. A sequence is an acyclic orientation of the Hamming graph K (n) xK (m) . Irreducible sequences in open shop are the local optimal elements. We present two variants of algorithms based on the specific properties of the H-comparability graph. The first is polynomial whereas the second is exponential. The irreducibility is co-NP. The stated properties argue whether it belongs to P. The complexity status of the considered decision problem is updated.