Digital Object Identifier (DOI) 10.1007/s10107-003-0411-9 Math. Program., Ser. B 98: 415–429 (2003) Rainer E. Burkard · Peter Butkoviˇ c Max algebra and the linear assignment problem Dedicated to Professor Egon Balas on the occasion of his 80th birthday. Received: May 31, 2002 / Accepted: January 27, 2003 Published online: March 28, 2003 – © Springer-Verlag 2003 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 assign- ment 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 re- alisation 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. Key words. max-algebra – assignment problem – permanent – regular matrix – discrete event system – characteristic maxpolynomial – best principal submatrix assignment problem – job rotation problem 1. Introduction In the max-algebra the conventional arithmetic operations of addition and multiplication in the real numbers are replaced by a ⊕ b := max(a, b), (1) a ⊗ b := a + b, (2) where a,b ∈ R := R ∪ {−∞}. The algebraic system ( R, ⊕, ⊗) offers an adequate language to describe problems from communication networks (Shimbel [23]), synchro- nization of production (Cuninghame-Green [8]) and transportation, shortest paths (e.g. Peteanu [21], Carr´ e [7], Gondran [17]) and discrete event systems, to mention just a few R.E. Burkard: Technische Universit¨ at Graz, Institut f¨ ur Mathematik, Steyrergasse 30, A-8010 Graz, Austria, e-mail: burkard@tugraz.at P. Butkovˇ c: School of Mathematics and Statistics, The University of Birmingham, Birmingham B15 2TT, UK, e-mail: p.butkovic@bham.ac.uk ∗ 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