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

AbstractLet twon×n matrices be given, namely a real matrixA=(aij) and a (0, 1)-matrixT=(tij). For a cyclic permutationσ=(i1,i2,...,ik) of a subset of N={1, 2, ..., n} we define μA;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)=maxσ μA;T(σ), the maximum cost-to-time ratio weight of the matricesA andT. Moreover a generalised eigenproblem is proposed.