Triangularization methods which have provided a tool for studying the production structure of an economy are considered. The previous triangularization methods are founded on a permutation theorem which defines a necessary condition to the triangularization problem: to permute the industries in an input–output matrix so as to maximize the sum of the below-diagonal elements. This paper shows that a sufficient condition can also be defined in a branch and bound state. Consequently, a powerful triangularization algorithm can be formulated.