Approximation algorithms for shop scheduling problems with minsum objective

Abstract

SUMMARYWe consider a general class of multiprocessor shop scheduling problems, preemptive or non-preemptive, 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 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 2e 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)|r ij ; dag j | S w S C S where S w S C S 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 C j ; F(P)|r j | w j C j and O C j . We also obtain a O(1) performance guarantee for the acyclic job shop problem J |p ij = 1; acyclic-dag j | S w S C S 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|r j | w j C j 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, non-preemptive, and no-wait) with m single-processor stages and total weighted job completion time objective.

Improved Bounds for Flow Shop Scheduling

“…Queyranne & Sviridenko [15] showed that an approximation algorithm for the above mentioned problems that produces a schedule with makespan a factor O(ρ) away from the lower bound lb can be used to obtain a O(ρ)-approximation algorithms for other objectives, including the sum of weighted completion times. The only known inapproximability result is by Hoogeveen, Schuurman & Woeginger [7], who showed that F || C j is NP-hard to approximate within a ratio better than 1 + for some small > 0.…”

From fluid relaxations to practical algorithms for job shop scheduling: the makespan objective

“…An interesting recent development is the work of Queyranne and Sviridenko (1999), in which they consider approximation algorithms for shop-scheduling problems with a minimal sum of job completion times objective. Their main result is the following: If there exists a polynomial-time algorithm for a class of multiprocessor job-shop problems that guarantees a makespan no larger than times the trivial lower bound (the so-called congestion-dilation bound), then they describe a polynomial-time algorithm for minimizing the weighted completion time that is within a factor of 8 of the optimum.…”

“…Their algorithm involves use of the approximation scheme for the makespan objective. Note that the polynomial-time approximation schemes for the makespan objective do not always satisfy the hypothesis of their statement: The work of Queyranne and Sviridenko (1999) requires a makespan guarantee that is within a factor of of a lower bound, not the optimal makespan itself. In fact, recent results of Hoogeveen et al (1998) show that the job-shop problem with the objective of minimizing weighted completion time does not have a polynomial-time approximation scheme unless P = NP .…”

Combinatorial Algorithms for Data Migration to Minimize Average Completion Time

“…Kim [15] proved that the problem is NP-hard when edge lengths are the same and showed that Graham's list scheduling algorithm [7], when guided by an optimal solution to a linear programming relaxation, gives an approximation ratio of 3. When edges have arbitrary lengths there are several constant factor approximation algorithms [5,15,20] with the best approximation guarantee being 5.03 [5].…”

“…This problem is a special case of the data migration problem [5]. Open shop scheduling problem has been studied in [3,12,19,20].…”