Solving Restricted Preemptive Scheduling on Parallel Machines with SAT and PMS

Restricted preemption plays a crucial role in reducing total completion time while controlling preemption overhead. A typical version of restricted preemptive models is k-restricted preemptive scheduling, where preemption is only allowed after a task has been continuously processed for at least k units of time. Though solving this problem of minimizing the makespan on parallel machines is NP-hard in general, it is of vital importance to obtain the optimal solution for small-sized problems, as well as for evaluation of heuristics. This paper proposes optimal strategies to the aforementioned problem. Motivated by the dramatic speed-up of Boolean Satisfiability (SAT) solvers, we make the first step towards a study of applying a SAT solver to the k-restricted scheduling problem. We set out to encode the scheduling problem into propositional Boolean logic and determine the optimal makespan by repeatedly calling an off-the-shelf SAT solver. Moreover, we move one step further by encoding the problem into Partial Maximum Satisfiability (PMS), which is an optimized version of SAT, so that the explicit successive calls of the solver can be eliminated. The optimal makespan of the problem and the performance of the proposed methods are studied experimentally. Furthermore, an existing heuristic algorithm is evaluated by the optimization methods.