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On the generation of circuits and minimal forbidden sets

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An Erratum to this article was published on 31 May 2004

Abstract.

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.

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References

  1. Alvarez-Valdés Olaguíbel, R., Tamarit Goerlich, J.M.: The project scheduling polyhedron: Dimension, facets, and liftig theorems. Eur. J. Oper. Res. 67, 204–220 (1993)

    Article  MATH  Google Scholar 

  2. Balas, E.: Project scheduling with resource constraints. In: Beale, E.M.L., (ed.), Applications of Mathematical Programming Techniques. The English University Press, London, 1970

  3. Bartusch, M.: An algorithm for generating all maximal independent subsets of posets. Computing 26, 343–354 (1981)

    MATH  MathSciNet  Google Scholar 

  4. Bartusch, M.: Optimierung von Netzplänen mit Anordnungsbeziehungen bei knappen Betriebsmitteln. PhD thesis, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany, 1984

  5. Bartusch, M., Möhring, R.H., Radermacher, F.J.: Scheduling project networks with resource constraints and time windows. Ann. Oper. Res. 16, 201–240 (1988)

    MathSciNet  Google Scholar 

  6. Boros, E., Elbassioni, K., Khachiyan, L., Makino, K.: Dual-bounded generating problems: All minimal integer solutions for a monotone system of linear inequalities. SIAM J. Comput. 31, 1624–1643 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boros, E., Gurvich, V., Hammer, P.L.: Dual subimplicants of positive boolean functions. Optim. Meth. Softw. 10, 147–156 (1998)

    MATH  MathSciNet  Google Scholar 

  8. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics. Philadelphia, PA, 1999

  9. Chvátal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Ann. Disc. Math. 1, 145–162 (1977)

    Article  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, 1979

  11. Igelmund, G., Radermacher, F.J.: Algorithmic approaches to preselective strategies for stochastic scheduling problems. Networks 13, 29–48 (1983)

    MATH  MathSciNet  Google Scholar 

  12. Igelmund, G., Radermacher, F.J.: Preselective strategies for the optimization of stochastic project networks under resource constraints. Networks 13, 1–28 (1983)

    MATH  MathSciNet  Google Scholar 

  13. Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Processing Lett. 27, 119–123 (1988)

    Article  MATH  Google Scholar 

  14. Kolisch, R., Sprecher, A.: PSPLIB - A project scheduling problem library. Eur. J. Oper. Res. 96, 205–216 (1996)

    Article  Google Scholar 

  15. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9, 558–565 (1980)

    MATH  MathSciNet  Google Scholar 

  16. Möhring, R.H.: Algorithmic aspects of comparability graphs and interval graphs. In: Rival, I., (ed.), Graphs and Order, NATO Advanced Science Institute Series, D. Reidel Publishing Company, Dordrecht, 1985, pp. 41–101

  17. Möhring, R.H., Skutella, M., Stork, F.: Scheduling with AND/OR precedence constraints. SIAM J. Comput. 2004. To appear

  18. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading, MA, 1994

  19. Pritsker, A.A.B., Watters, L.J., Wolfe, P.M.: Multi project scheduling with limited resources: A zero-one programming approach. Manage. Sci. 16, 93–108 (1969)

    Article  Google Scholar 

  20. Provan, J.S., Ball, M.O.: The complexity of counting cuts and of the probability that a graph is connected. SIAM J. Comput. 12, 777–788 (1983)

    MATH  MathSciNet  Google Scholar 

  21. PSPLIB. http://www.bwl.uni-kiel.de/Prod/psplib/, 2000

  22. Radermacher, F.J.: Scheduling of project networks. Ann. Oper. Res. 4, 227–252 (1985)

    MATH  MathSciNet  Google Scholar 

  23. Roy, B., Sussmann, B.: Les problèmes d’ordonnancement avec contraintes disjonctives. Note D.S. 9 bis, SEMA, Paris, France, 1964

  24. Stork, F.: Stochastic Resource-Constrained Project Scheduling. PhD thesis, Institut für Mathematik, Technische Universität Berlin, Berlin, Germany, 2001

  25. Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  26. Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410–421 (1979)

    MATH  MathSciNet  Google Scholar 

  27. Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebraic Disc. Meth. 3, 351–358 (1982)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. ILOG Deutschland GmbH, Ober-Eschbacher Straße 109, D-61352, Bad Homburg, Germany

    Frederik Stork

  2. Universiteit Maastricht, Faculty of Economics and Business Administration, Quantitative Economics, P.O. Box 616, NL-6200, MD, Maastricht, The Netherlands

    Marc Uetz

Authors
  1. Frederik Stork
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  2. Marc Uetz
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Correspondence to Marc Uetz.

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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.

An erratum to this article is available at http://dx.doi.org/10.1007/s10107-004-0532-9.

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Stork, F., Uetz, M. On the generation of circuits and minimal forbidden sets. Math. Program. 102, 185–203 (2005). https://doi.org/10.1007/s10107-004-0512-0

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