Jan Karel Lenstra

Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.

Job shop scheduling by simulated annealing

We describe an approximation algorithm for the problem of finding the minimum makespan in a job shop. The algorithm is based on simulated annealing, a generalization of the well known iterative improvement approach to combinatorial optimization problems. The generalization involves the acceptance of cost-increasing transitions with a nonzero probability to avoid getting stuck in local minima. We prove that our algorithm asymptotically converges in probability to a globally minimal solution, despite the fact that the Markov chains generated by the algorithm are generally not irreducible. Computational experiments show that our algorithm can find shorter makespans than two recent approximation approaches that are more tailored to the job shop scheduling problem. This is, however, at the cost of large running times.

Scheduling subject to resource constraints: classification and complexity

Abstract In deterministic sequencing and scheduling problems, jobs are to be processed on machines of limited capacity. We consider an extension of this class of problems, in which the jobs require the use of additional scarce resources during their execution. A classification scheme for resource constraints is proposed and the computational complexity of the extended problem class is investigated in terms of this classification. Models involving parallel machines, unit-time jobs and the maximum completion time criterion are studied in detail; other models are briefly discussed.

Complexity of Scheduling under Precedence Constraints

Precedence constraints between jobs that have to be respected in every feasible schedule generally increase the computational complexity of a scheduling problem. Occasionally, their introduction may turn a problem that is solvable within polynomial time into an NP-complete one, for which a good algorithm is highly unlikely to exist. We illustrate the use of these concepts by extending some typical NP-completeness results and simplifying their correctness proofs for scheduling problems involving precedence constraints.

Approximation algorithms for scheduling unrelated parallel machines

We consider the following scheduling problem. There are m parallel machines and n independent jobs. Each job is to be assigned to one of the machines. The processing of job j on machine i requires time pij. The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unless P = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

Chapter 9 Sequencing and scheduling: Algorithms and complexity

Publisher Summary This chapter discusses different types of sequencing and scheduling problems, and describes different types of algorithms and the concepts of complexity theory. A class of deterministic machine scheduling problems has been introduced in the chapter. The chapter also deals with the single machine, parallel machine and multi-operation problems in this class, respectively. The two generalizations of the deterministic machine-scheduling model have been presented in the chapter. A deterministic scheduling model may give rise to various stochastic counterparts, as there is a choice in the parameters that are randomized, in their distributions, and in the classes of policies that can be applied. A characteristic feature of these models is that the stochastic parameters are regarded as independent random variables with a given distribution and that their realization occurs only after the scheduling decision has been made. In the deterministic model, one has perfect information, and capitalizing on it in minimizing the realization of a performance measure may require exponential time.

Recent Developments in Deterministic Sequencing and Scheduling: A Survey

The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. We survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. This paper is a revised version of the survey by Graham et al. (Ann. Discrete Math. 5(1979) 287–326) , with emphasis on recent developments.

Generating All Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms

Suppose that an independence system

A General Bounding Scheme for the Permutation Flow-Shop Problem

(E,\mathcal {I})

Minimizing Total Costs in One-Machine Scheduling

is characterized by a subroutine which indicates in unit time whether or not a given subset of E is independent. It is shown that there is no algorithm for generating all the K maximal independent sets of such an independence system in time polynomial in