The purpose of this note is to propose a complexity framework for the analysis of high multiplicity scheduling problems. Part of this framework relies on earlier work aiming at the definition of outputsensitive complexity measures for the analysis of algorithms which produce "large" outputs. However, different classes emerge according as we look at schedules as sets of starting times, or as related single-valued mappings.

In this paper, a graph partitioning problem that arises in the design of SONET/SDH networks is defined and formalized. Approximation algorithms with performance guarantees are presented. To solve this problem efficiently in practice, fast greedy algorithms and a tabu-search method are proposed and analyzed by means of an experimental study.

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This note revisits the maximum deviation just-in-time (MDJIT) scheduling problem previously investigated by Steiner and Yeomans (1993). Its main result is a set of algebraic necessary and su±cient conditions for the existence of a MDJIT schedule with a given objective function value. These conditions are used to provide a¯ner analysis of the complexity of the MDJIT problem. The note also investigates various special cases of the MDJIT problem and suggests several questions for further investigation. Acknowledgements. The authors are grateful to Alexandre Grigoriev, Joris van de Klundert and Andr ¶ as Seb} o for several useful discussions regarding the topic of this note, and to Gerhard Woeginger for pointing out the relevance of balanced sequences.

We consider a robotic cell, consisting of a fiow-shop in which the machines are served by a single central robot. We concentrate on the case where only one part type is produced and want to analyze the conjecture of Sethi, Sriskandarajah, Sorger, Blazewicz and Kubiak, This well-known conjecture claims that the repetition of the best one-unit production cycle will yield the maximum throughput rate in the set of all possible robot moves. The conjecture holds for two and three machines, but the existing proof by van de Klundert and Crama for the three-machine case is extremely tedious.We adopt the theoretical background developed by Crama and van de Klundert, Using a particular state graph, the fc-unit production cycles are represented as special paths and cycles for which general properties and bounds for the m-machine robotic cell can be obtained. By means of these concepts, we shall give a concise proof for the validity of the conjecture for the three-machine case, RESUMEOn considere une cellule de production composee de m machines et d'un robot charge du transfert des pieces entre Ies machines, Les pieces a produire sont toutes identiques et doivent passer successivement, et dans le meme ordre, sur toutes les machines, tjne machine ne peut accueillir plus d'une piece a la fois. On souhaite analyser la conjecture de Sethi, Sriskandarajah, Sorger, Blazewicz et Kubiak qui pretend que le taux maximum de production peut etre obtenu en repetant le meilleur cycle de production d'une seule piece, Cette conjecture a ete demontree pour deux et trois machines, mais la preuve de van de Klundert et Crama pour le cas de trois machines est tres fastidieuse.En utilisant les bases theoriques introduites par Crama et van de Klundert, nous definissons des graphes particuliers, les graphes d'etat, dans lesquels les cycles de production de k pieces sont representes par des chemins et par des cycles, Ces graphes permettent de trouver des proprietes et des bornes pour les cellules de production, Au moyen de ces concepts, nous donnons une preuve concise de la validite de la conjecture pour le cas de trois machines.