Finding all essential terms of a characteristic maxpolynomial

Authors:
Rainer E. Burkard;Peter Butkovič
Affiliations:
Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria;School of Mathematics and Statistics, The University of Birmingham, Birmingham B15 2TT, UK
Venue:
Discrete Applied Mathematics
Year:
2003

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Abstract

Let us denote a ⊕ b = max(a,b) and a ⊗ b = a + b for a, b ∈ oR = R ∪ {-∞} and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n2(m + n log n)) algorithm for finding all essential terms of the max-algebraic characteristic polynomial of an n × n matrix over oR with m finite elements. In the cases when all terms are essential, this algorithm also solves the following problem: Given an n × n matrix A and k ∈ {1,...,n}, find a k × k principal submatrix of A whose assignment problem value is maximum.