Regularity of matrices in min-algebra and its time- complexity Original Research Article

Let ¢G = (G, ⊗, ⩽) be a linearly ordered, commutative group and ⊕ be defined by a ⊕ b = min(a, b) for all a, b ϵ G. Extend ⊕, ⊗ to matrices and vectors as in conventional linear algebra. An n × n matrix A with columns A1,…,An is called regular if ∑jϵU⊕ λj ⊗ Aj = ∑jϵV⊕λj ⊗ Aj does not hold for any λ1,…,λn ϵ G, σ ≠ U, V ⊆ {1, 2,…, n}, U ∩ V = σ. We show that the problem of checking regularity is polynomially equivalent to the even cycle problem. We also present two other types of regularity which can be checked