A. H. G. Rinnooy Kan

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.

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 vehicle routing and scheduling problems

The complexity of a class of vehicle routing and scheduling problems is investigated. We review known NP-hardness results and compile the results on the worst-case performance of approximation algorithms. Some directions for future research are suggested. The presentation is based on two discussion sessions during the Workshop to Investigate Future Directions in Routing and Scheduling of Vehicles and Crews, held at the University of Maryland at College Park, June 4–6, 1979.

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.

The complexity of the network design problem

In the network design problem we are given a weighted undirected graph. We wish to find a subgraph which connects all the original vertices and minimizes the sum of the shortest path weights between all vertex pairs, subject to a budget constraint on the sum of its edge weights. In this note we establish NP-completeness for the network design problem, even for the simple case where all edge weights are equal and the budget restricts the choice to spanning trees. This result justifies the development of enumerative optimization methods and of approximation algorithms, such as those described in a recent paper by R. Dionne and M. Florian.

Bounds and Heuristics for Capacitated Routing Problems

In a capacitated routing problem, the objective is to minimize the total distance travelled by vehicles of limited capacity to serve a set of customers that are located in the Euclidean plane. We develop asymptotically optimal bounds and heuristics for this problem, under the assumption that the capacity of a vehicle is expressed in terms of an upper bound on the number of customers that it can serve. The analysis culminates in an algorithm that, for a given capacity and given (epsilon), will find a solution with relative error at most (epsilon) in time polynomial in the number of customers.

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.

A stochastic method for global optimization

A stochastic method for global optimization is described and evaluated. The method involves a combination of sampling, clustering and local search, and terminates with a range of confidence intervals on the value of the global optimum. Computational results on standard test functions are included as well.

Computational complexity of discrete optimization problems

Recent developments in the theory of computational complexity as applied to combinatorial problems have revealed the existence of a large class of so-called NP-complete problems, either all or none of which are solvable in polynomial time. Since many infamous combinatorial problems have been proved to be NP-complete, the latter alternative seems far more likely. In that sense, NP-completeness of a problem justifies the use of enumerative optimization methods and of approximation algorithms. In this paper we give an informal introduction to the theory of NP-completeness and derive some fundamental results, in the hope of stimulating further use of this valuable analytical tool.

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

Suppose that an independence system