Approximation algorithms for shop scheduling problems with minsum objective: A correction

We 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 class of objective functions including weighted sums of job, operations and stage completion times. We present a general approximation method combining a linear programming relaxation in the operation completion times, with any algorithm for the makespan version of these problems without release dates. If the latter produces a schedule with makespan no larger than ρ times the “trivial lower bound” consisting of the largest of all stage average loads (or “congestion”) and job lengths (or “dilation”), then our method produces a feasible schedule with minsum objective no larger than 2eρ times the optimum where 2e ≈ 5.44. This leads in particular to a polynomial time algorithm with polylogarithmic performance guarantee for the minsum multiprocessor dag-shop problem J(P )|rij , dagj | ∑ S wSCS where ∑ S wSCS is a general minsum objective including weighted sum of operation and job completion times, stages makespans and others, whereas the best known earlier performance guarantees were O(m) (where m is the number of stages) for the special cases J ||Cj (Gonzalez and Sahni, 1978), F (P )|rj | ∑ wjCj (Schulz, 1996) and O||Cj (Achugbue and Chin, 1982). We also obtain a O(1) performance guarantee for the acyclic job shop problem J |pij = 1, acyclic− dagj| ∑ S wSCS with unit processing times and weighted sum of operation (or job) completion time objective. Our results extend to a broad class of minsum objective functions including some convex objectives related to load balancing. We then present an improved 5.83-approximation algorithm for the open shop problem O|rj | ∑ wjCj with total weighted job completion time objective. We conclude with a very simple method which yields O(m)-approximation algorithms for various job shop problems (preemptive, nonpreemptive, and no-wait) with m single-processor stages and total weighted job completion time objective.