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Combinatorial Optimization & Graph Algorithms group (COGA)Basic Sequencing

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The problems considered here as basic sequencing problems are basically classic machine scheduling problems which reduce to finding a minimum cost sequence of jobs. In the considered setting, the cost incurred by a job solely depends on the neighboring jobs in the sequence or the job's completion time. We investigate the following problems:

This is joint work with Tobias Jacobs (previously in this Priority Programme, later at NII Tokyo). The work on Eulerian extensions is also joint work with Nicole Megow (MPII Saarbrücken).

Generalized Min-Sum-Scheduling

We consider an offline single machine scheduling problem with generalized min-sum objective. In contrast to the classic variant where one aims for minimizing the (weighted) average of completion times or flow times, here some nonlinear cost function is applied to the values beforehand. The choice of  cost function allows to model several objectives. By utilizing Lp-norms (or closely related monomials), one can flexibly compromise between average and worst-case cost. Moreover, the cost function can be interpreted as nonuniform processor speed.

Formally, given a set of n jobs with processing times pj and weight wj for j=1, ..., n, one aims for a sequence of the jobs minimizing ∑j wj g(Cj) where g is some non-decreasing cost function and Cj denotes the completion time of job j.

We show that for piecewise linear  cost functions the problem is strongly NP-hard while it is in P for exponential cost functions. The main theoretical contribution is a tight analysis of the approximation guarantee of Smith's rule (a.k.a. Highest Density First) under any particular convex or concave cost function. In addition, we evaluate the optimality gap of Smith's rule empirically for different monomial cost functions. To overcome unrealistic worst case instances, we also give tight approximation bounds that are parametrized by the minimum, maximum, and total processing time. As the approximation factors tend to be very small, they yield valuable lower bounds on the optimum that can utilized in exact solution methods.

For quadratic cost functions, throughout the past decades, great effort has been made to develop fast exact algorithms. The efficiency of these methods heavily depends on the utilization of structural properties of optimal schedules such as order constraints, i.e., sufficient conditions for pairs of jobs to appear in a certain order. We enhance the map of known order constraints by proving an extended version of a constraint that has been conjectured by Mondal and Sen (European Journal of Operational Research 125, 2000). Besides proving this conjecture, our main contribution is an extensive experimental study where the influence of different kinds order constraints on the performance of exact algorithms is systematically evaluated. In addition to a best-first graph search algorithm, we test a Quadratic Integer Programming formulation that admits to add order constraints as additional linear constraints.

Our experiments are based on sets of  problem instances that have been generated using  a new method which allows us to adjust a certain degree of difficulty of the instances. They are available for download.

Eulerian Extension Problems

We propose a new technique for considering the complexity of certain sequencing problems like the Gilmore-Gomory type Traveling Salesman Problem (GG-TSP) or the new flowshop problem we investigate in our corresponding publication. These problems have a natural interpretation as Eulerian Extension Problems. Briefly, in this approach for an instance of a sequencing problem, we build a Eulerian graph in which every tour corresponds to an optimal solution and vice versa. The main advantage of this approach is that instead of only one optimal solution, we obtain the entire set of optimal solutions, improving in that sense the original algorithm for the GG-TSP by Gilmore and Gomory (Operations Research, 1964).

The new objective function which we consider for no-wait flowshop scheduling is motivated by the process of strand casting in steel manufacturing. Since idle-times on the last machine, the strand casting machine, cause immense setup times, we aim to minimize the total number of idle times on the last machine in the process.

Production site for steel manufacturing.

Using the Eulerian Extension approach, we show that in case of only single-machine stages, this problem is solvable in polynomial time for two processing stages, and it is strongly NP-hard for more processing stages. Furthermore, we show that the problem can still be solved optimally when there are two machine stages and one machine in the first stage. For all other cases we show the strong NP-hardness, thus completely resolving the complexity status of this problem.

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On the performance of Smith's rule in single-machine scheduling with nonlinear cost
Citation key HoehnJacobs2012b
Author Höhn, Wiebke and Jacobs, Tobias
Title of Book Proc. of the 10th Latin American Theoretical Informatics Symposium (LATIN)
Pages 482-493
Year 2012
DOI 10.1007/978-3-642-29344-3_41
Volume 7256
Publisher Springer
Series LNCS
Abstract We consider the problem of scheduling jobs on a single machine. Given some continuous cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each individual job is determined by the cost function value at the job's completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation factor of Smith's rule under any particular convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. To overcome unrealistic worst case instances, we also give tight bounds that are parameterized by the minimum, maximum, and total processing time.
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