Abstract
Many programming models, e.g., MapReduce, introduce precedence constraints between the jobs. This paper formalizes a notion of precedence constraints, called fractional precedence constraints, where the progress of follower jobs only has to lag behind (fractionally) their leads. For a general set of fractional precedence constraints between the jobs, this paper provides a new class of preemptive scheduling algorithms on unrelated machines that have arbitrary processing speeds. In particular, for a given makespan, we establish both sufficient and necessary conditions on the existence of a feasible job schedule, and then propose an efficient scheduling algorithm based on a novel matrix decomposition method, if the sufficient conditions are satisfied. The algorithm is shown to be a Polynomial-Time Approximation Scheme (PTAS), i.e., its solution is able to achieve any feasible makespan with an approximation bound of 1 + epsilon, for an arbitrary epsilon > 0. (C) 2021 Elsevier Inc. All rights reserved.