New and improved algorithms for minsum shop scheduling

Abstract

Translate Article

We consider a general class of multiprocessor shop scheduling problems with a reinsure objective, and present approximation methods based on linear programming relaxations in the operation completion times. These LP relaxations use new classes of valid inequalities for multistage jobs. We first consider open shop problems with total weighted job completion time objective. For the nonpreemptive problem O[[ ~ wj Cj, we introduce LP-based precedence and derive a 5.83-approximation algorithm. For its preemptive version, O[pmtn[ y~wjCj, we show that a simple job-based greedy algorithm, using directly the LP solution, yields a 3o approximation. We then consider a general class of multiprocessor shop scheduling problems, preemptive or nonpreemptive, with precedence constraints between operations, with job or operation release dates, and with a general reinsure objective. This class of objectives includes, among others, weighted sums of operations completion times, job completion times, stage mal~espans and the overall makespan. We combine the LP relaxation with two known techniques: (i) partitioning the set of operations using time intervals with geometrically increasing lengths; and (ii) an approximation algorithm for the makespan version of these problems without release dates. When the latter produces a schedule with makespan no larger than p times the trivial lower consisting of the largest of all stage average loads (or congestion?) and job lengths (or dilation), we obtain a 2epapproximation, where 2 e < 5.44. This leads in particular to a polylogarithmic approximation for the reinsure multiprocessor dag-shop problem, and a O(1) approximation for the reinsure, acyclic job shop problem with unit processing times. These performance guarantees also bound the gap between optimum preemptive and nonpreernptive schedules