Decision diagram-based integer programming for the paired job scheduling problem

The paired job scheduling problem seeks to schedule n jobs on a single machine, each job consisting of two tasks for which there is a mandatory minimum waiting time between the completion of the first task and the start of the second task. We provide complexity results for problems defined by three commonly used objective functions. We propose an integer programming formulation based on a decision diagram decomposition that models the objective function and some of the challenging constraints in the space of the flow variables stemming from the diagrams while enforcing the simpler constraints in the space of the original scheduling variables. We then show how to simplify our reformulation by projecting out a subset of the flow variables, resulting in a lifted reformulation for the problem that can be obtained without building the decision diagrams. Computational results show that our proposed model performs considerably better than a standard time-indexed formulation over a set of randomly generated instances.