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Powers of matrices over an extremal algebra with applications to periodic graphs

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Abstract

Consider the extremal algebra

=(ℝ∪{∞},min,+), using + and min instead of addition and multiplication. This extremal algebra has been successfully applied to a lot of scheduling problems. In this paper the behavior of the powers of a matrix over

is studied. The main result is a representation of the complete sequence (A m) m∈ℕ which can be computed within polynomial time complexity. In the second part we apply this result to compute a minimum cost path in a 1-dimensional periodic graph.

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References

  1. Bacceli F, Cohen G, Olsder G, Quadrat J (1992) Synchronization and linearity. J. Wiley and Sons

  2. Backes W, Schwiegelshohn U, Thiele L (1992) Analysis of free schedule in period graphs. Proceedings of Symposium on Parallel Algorithms and Architectures: 333–343

  3. Bellman R, Karush W (1961) A new functional transform in analysis: The maximum transform. Bull. Am. Math. Soc. 67:501–503

    Google Scholar 

  4. Carré B (1979) Graphs and networks. Oxford

  5. Cohen E, Megiddo N (1993) Strongly polynomial-time and nc algorithms for detecting cycles in periodic graphs. Journal of the Association for Computing Machinery 40:791–830

    Google Scholar 

  6. Cohen G, Dubois D, Quadrat J, Viot M (1985) A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Transactions on Automatic Control, AC-30: 210–220

    Google Scholar 

  7. Cuninghame-Green R (1979) Minimax-algebra. Lecture Notes in economics and mathematical systems 166, Springer Verlag, Berlin, New York

    Google Scholar 

  8. Gondran M, Minoux M (1984) Graphs and algorithms. J. Wiley & Sons

  9. Höfling F, Wanke E (1995) Polynomial algorithms for minimum cost paths in periodic graphs. SIAM Journal on Computing: 1051–1067

  10. Iwano K, Steiglitz K (1987) Testing for cycles in infinite graphs with periodic structure. Proceedings of Annual ACM Symposium on Theory of Computing: 46–55

  11. Karp R (1978) A characterization of the minimum cycle mean in a digraph. Discrete Mathematics 23:309–311

    Google Scholar 

  12. Karp R, Miller R, Winograd S (1967) The organization of computations for uniform recurrence equations. J. ACM 2:563–590

    Google Scholar 

  13. Orlin JB (1984) Some problems on dynamic/periodic graphs. In: Progress in Combinatorial Optimization: 273–293

  14. Worobjov N (1967) Extremal algebra of positive matrices. (in Russian), Elektronische Informationsverarbeitung und Kybernetik

  15. Zimmermann U (1981) Linear and combinatorial optimization in ordered algebraic structures. Annals of discrete mathematics 10, North Holland

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  1. Institut für Flugführung, DLR, Lilienthalplatz 7, 38108, Braunschweig, Germany

    Karl Nachtigall

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  1. Karl Nachtigall
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Nachtigall, K. Powers of matrices over an extremal algebra with applications to periodic graphs. Mathematical Methods of Operations Research 46, 87–102 (1997). https://doi.org/10.1007/BF01199464

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  • DOI: https://doi.org/10.1007/BF01199464

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