TheO(n³) algorithm for a special case of the maximum cost-to-time ratio cycle problem and its coherence with an eigenproblem of a matrix

Abstract

Let twon×n matrices be given, namely a real matrixA=(a_ij) and a (0, 1)-matrixT=(t_ij). For a cyclic permutationσ=(i₁,i₂,...,i_k) of a subset of N={1, 2, ..., n} we define μ_A;T(σ), the cost-to-time ratio weight ofσ, as\((a_{i_1 i_2 } + \cdots + a_{i_k i_1 } )/(t_{i_1 i_2 } + \cdots + t_{i_k i_1 } )\). This paper presents an O(n³) algorithm for finding λ(A;T)=max_σ μ_A;T(σ), the maximum cost-to-time ratio weight of the matricesA andT. Moreover a generalised eigenproblem is proposed.