We give a state-of-the-art survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modi®cations of a basic heuristic and investigate their usefulness for evaluating the thickness and determining a decomposition of a graph in planar subgraphs.

We present a new average case analysis for the problem of scheduling n jobs on m machines so that the sum of job completion times is minimized. Our goal is to use the concept of competitive ratio-which is a typical worst case notion-also within an average case analysis. We show that the classic SEPT scheduling strategy with (n) worst-case competitive ratio achieves an average of O(1) under several natural distributions, among them the exponential distribution. Our analysis technique allows to also roughly estimate the probability distribution of the competitive ratio. Thus, our result bridges the gap between worst case and average case performance guarantee.

Scheduling problems of minimizing the makespan on identical parallel machines are among the most well‐studied problems — especially in the field of approximation. In modern industrial software however, it has become standard to work on a variant of this problem, where some of the jobs are already fixed in the schedule. The remaining jobs are to be assigned to the machines in such a way that they do not overlap with fixed jobs. This problem variant is the root of many real‐world scheduling problems where pre‐assignments on the machines are considered, such as cleaning times or jobs that have already started. In our paper we investigate the approximability of the scheduling problem with fixed jobs. We present a polynomial‐time approximation scheme (PTAS) for the case that the number m of machines is constant. For our PTAS we propose a new technique by partitioning an underlying packing problem into a reasonable unrelated family of restricted bin packing problems. We also generalize the PTAS to the case that the machines are independent and run at different speeds. Moreover, we will demonstrate that, assuming P≠NP, there is no arbitrarily close approximation in the general case when the number of machines is part of the input. This will be extended by showing that there is no asymptotic PTAS in the general machine case. We finally show that there exists no FPTAS in the constant machine case, unless P=NP. These results contrast to the classical problem of minimizing the makespan where the existence of a PTAS resp. of an FPTAS for the variable resp. the constant machine case has been proven. Copyright © 1999 John Wiley & Sons, Ltd.

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Scheduling problems of minimizing the makespan on identical parallel machines are among the most wellstudied problems -especially in the ÿeld of approximation. In modern industrial software however, it has become standard to work on a variant of this problem, where some of the jobs are already ÿxed in the schedule. The remaining jobs are to be assigned to the machines in such a way that they do not overlap with ÿxed jobs. This problem variant is the root of many real-world scheduling problems where pre-assignments on the machines are considered, such as cleaning times or jobs that have already started. In our paper we investigate the approximability of the scheduling problem with ÿxed jobs. We present a polynomial-time approximation scheme (PTAS) for the case that the number m of machines is constant. For our PTAS we propose a new technique by partitioning an underlying packing problem into a reasonable unrelated family of restricted bin packing problems. We also generalize the PTAS to the case that the machines are independent and run at di erent speeds. Moreover, we will demonstrate that, assuming P = NP, there is no arbitrarily close approximation in the general case when the number of machines is part of the input. This will be extended by showing that there is no asymptotic PTAS in the general machine case. We ÿnally show that there exists no FPTAS in the constant machine case, unless P=NP. These results contrast to the classical problem of minimizing the makespan where the existence of a PTAS resp. of an FPTAS for the variable resp. the constant machine case has been proven.