Eugene L. Lawler

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.

Branch-and-Bound Methods: A Survey

The essential features of the branch-and-bound approach to constrained optimization are described, and several specific applications are reviewed. These include integer linear programming Land-Doig and Balas methods, nonlinear programming minimization of nonconvex objective functions, the traveling-salesman problem Eastman and Little, et al. methods, and the quadratic assignment problem Gilmore and Lawler methods. Computational considerations, including trade-offs between length of computation and storage requirements, are discussed and a comparison with dynamic programming is made. Various applications outside the domain of mathematical programming are also mentioned.

The Recognition of Series Parallel Digraphs

We present a linear-time algorithm to recognize the class of vertex series-parallel (VSP) digraphs. Our method is based on the relationship between VSP digraphs and the class of edge series-parallel multidigraphs. As a byproduct of our analysis, we obtain efficient methods to compute the transitive closure and transitive reduction of VSP digraphs, and to test isomorphism of minimal VSP digraphs.

A “Pseudopolynomial” Algorithm for Sequencing Jobs to Minimize Total Tardiness

Suppose n jobs are to be processed by a single machine. Associated with each job j are a fixed integer processing time pi, a due date di, and a positive weight wj. The weighted tardiness of job j in a given sequence is wj, max (O, Ci, - di), where Ci is the completion time of job j. Assume that the weighting of jobs is “agreeable”, in the sense that pi

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.

Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints

Suppose n jobs are to be sequenced for processing by a single machine, with the object of minimizing total weighted completion time. It is shown that the problem is NP-complete if there are arbitrary precedence constraints. However, if precedence constraints are “series parallel”, the problem can be solved in O( n log n ) time. This result generalizes previous results for the more special case of rooted trees. It is also shown how a decomposition procedure suggested by Sidney can be implemented in polynomial-bounded time. Equivalence of the sequencing problem with the optimal linear ordering problem for directed graphs is discussed.

On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming

It IS shown that certain problems of optimal preemptive scheduling of unrelated parallel processors can be formulated and solved as hnear programming problems As a by-product of the linear programming formulaUons of these problems, upper bounds are obtained on the number of preempuons required for optimal schedules In particular it is shown that no more than O(m 2) preemptions are necessary, m order to schedule n jobs on m unrelated processors so as to minimize makespan

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

Suppose that an independence system

Linear-time computation of optimal subgraphs of decomposable graphs

(E,\mathcal {I})

Preemptive scheduling of uniform machines subject to release dates

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