Introduction to max-linear programming

Let a ⊕ b =  max(a, b) and a ⊗ b = a + b for a, b ∈ ℝ. Extend this pair of operations to matrices and vectors in the same way as in linear algebra. Being motivated by scheduling of multiprocessor interactive systems, we introduce max-linear programs of the form f^T ⊗ x → min (or max) subject to A ⊗ x ⊕ c = B ⊗ x ⊕ d and develop solution methods for both of them. We prove that these methods are pseudo-polynomial if all entries are integers. This result is based on an existing pseudo-polynomial algorithm for solving the systems of the form A ⊗ x = B ⊗ y.