On Machine Dependency in Shop Scheduling

One of the main restrictions in scheduling problems are the machine (resource) restrictions: each machine can perform at most one job at a time. We explore machine dependencies is shop scheduling problems representing them as graphs. Our study shows that the structure of machine dependency graphs is important in the complexity analysis of shop scheduling problems. We call acyclic a shop scheduling problem with the acyclic machine dependency graph. Here we first consider the periodic jobshop scheduling problem and present a linear-time algorithm for its subclass which machine dependency graphs are allowed to have only simple so-called parti-colored cycles. This result is tight as the trivial extension of the problem becomes NP-hard. We show that the acyclic preemptive open-shop problem with at most m− 2 preemptions can be solved in linear time. This result is also tight as the same problem with at most m − 3 preemptions is NP-hard. We also show that very simple acyclic shop scheduling problems are NP-hard; for example, any flow-shop with a single job with three operations and the rest of the jobs with a single non-zero length operation is