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.