On the description of a class of problems solvable by a coordinate-wise ascent algorithm

In an earlier paper we obtained sufficient conditions for the solvability of some integer programming problems by a coordinatewise-ascent algorithm. In the case of the specification of the set of admissible solutions of the problem by systems of linear inequalities with nonnegative integer coefficients, these conditions can be expressed in terms of the properties of some families of sets closely associated with the structure of the system of linear constraints and with the objective function of the problem. In the present paper we give a more complete description (characterization) of these families of sets, based on a special form of representability of these families by parallel-sequential networks.