Regularity of matrices in min-algebra and its time- complexity

Abstract

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 A 1,…, A n is called regular if ∑ jϵU ⊕ λ j ⊗ A j = ∑ jϵV ⊕ λ j ⊗ A j does not hold for any λ 1,…, λ n 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 in O( n 3) operations.