On the generation of circuits and minimal forbidden sets

We present several complexity results related to generation and counting of all circuits of an independence system. Our motivation to study these problems is their relevance in the solution of resource constrained scheduling problems, where an independence system arises as the subsets of jobs that may be scheduled simultaneously. We are interested in the circuits of this system, the so-called minimal forbidden sets, which are minimal subsets of jobs that must not be scheduled simultaneously. As a consequence of the complexity results for general independence systems, we obtain several complexity results in the context of resource constrained scheduling. On that account, we propose and analyze a simple backtracking algorithm that generates all minimal forbidden sets for such problems. The performance of this algorithm, in comparison to a previously suggested divide-and-conquer approach, is evaluated empirically using instances from the project scheduling library PSPLIB.Acknowledgement We appreciate the input of two anonymous referees. Particularly the deep remarks of one of them greatly improved our understanding of several issues; he also suggested the simplified Example 1. We thank Marc Pfetsch and Alexander Grigoriev for several enlightening discussions. Marc Pfetsch also pointed us to the paper 15]. Parts of this work were done while the authors were PhD students at the Technische Universität Berlin, Germany, where Frederik Stork was supported by DFG grant Mo 446/3-3, and Marc Uetz was supported by bmb+f grant 03-MO7TU1-3 and GIF grant I 246-304.02/97.