New complexity results for shop scheduling problems with agreement graphs

We investigate the problem of scheduling jobs on open shop and flow shop systems of two machines, subject to the constraints that only some jobs can be scheduled simultaneously on different machines. These constraints are given by a simple undirected graph G, called the agreement graph. The objective is to minimize the makespan. We first show that the open shop problem with two distinct values of release times is NP-hard even for operation sizes in {1, 2} and we present a restricted case that can be solved in polynomial time. Then, we consider the problem with null release times and we show that when restricted to arbitrary trees, the problem is NP-hard. We also present a linear algorithm when restricted to caterpillars and we show that this algorithm can be used to solve the case of cycles in polynomial time. For the flow shop problem, we show that the problem is NP-hard for trees and that this result holds even if the preemption is allowed. (C) 2021 Elsevier B.V. All rights reserved.