On totally dual integral systems

Let A be a rational (m x n)-matrix and b be a rational m-vector. The linear system Ax< b is said to be totally dual integral (TDI) if for all integer n-vectors c, the linear program min{b’v: A’y= c; yr 0} has an integer-valued optimum solution if it has an optimum solution. The contents of this paper can be divided into three parts: First of all an attempt is made to characterize special classes of TDI systems. These TDI systems are classified on the basis of existence of totally unimodular active matrix sets of specific types. Secondly, a composition scheme for generating TDI systems is considered. This contains as a special case the Edmonds-Giles system. The power of this scheme is fully exploited. Finally a new TDI system is introduced which contains as special cases certain interesting old and new TDI systems.