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Max algebra and the linear assignment problem

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Abstract.

Max-algebra, where the classical arithmetic operations of addition and multiplication are replaced by a⊕b:=max(a, b) and a⊗b:=a+b offers an attractive way for modelling discrete event systems and optimization problems in production and transportation. Moreover, it shows a strong similarity to classical linear algebra: for instance, it allows a consideration of linear equation systems and the eigenvalue problem. The max-algebraic permanent of a matrix A corresponds to the maximum value of the classical linear assignment problem with cost matrix A. The analogue of van der Waerden's conjecture in max-algebra is proved. Moreover the role of the linear assignment problem in max-algebra is elaborated, in particular with respect to the uniqueness of solutions of linear equation systems, regularity of matrices and the minimal-dimensional realisation of discrete event systems. Further, the eigenvalue problem in max-algebra is discussed. It is intimately related to the best principal submatrix problem which is finally investigated: Given an integer k, 1≤k≤n, find a (k×k) principal submatrix of the given (n×n) matrix which yields among all principal submatrices of the same size the maximum (minimum) value for an assignment. For k=1,2,...,n, the maximum assignment problem values of the principal (k×k) submatrices are the coefficients of the max-algebraic characteristic polynomial of the matrix for A. This problem can be used to model job rotations.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Rainer E. Burkard

  2. School of Mathematics and Statistics, The University of Birmingham, Birmingham, B15 2TT, UK

    Peter Butkovič

Authors
  1. Rainer E. Burkard
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  2. Peter Butkovič
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Corresponding author

Correspondence to Rainer E. Burkard.

Additional information

This research has been supported by the Engineering and Physical Sciences Research Council grant RRAH07961 ``Unresolved Variants of the Assignment Problem'' and by the Spezialforschungsbereich F 003 ``Optimierung und Kontrolle'', Projektbereich Diskrete Optimierung.

Mathematics Subject Classification (2000): 90C27, 15A15, 93C83

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Burkard, R., Butkovič, P. Max algebra and the linear assignment problem. Math. Program., Ser. B 98, 415–429 (2003). https://doi.org/10.1007/s10107-003-0411-9

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