A Graphical Theorem of the Alternative for UTVPI Constraints

There exist quite a few theorems of the alternative for linear systems in the literature, with Farkas' lemma being the most famous. All these theorems have the following form: We are given two closely related linear systems such that one and exactly one has a solution. Some specialized classes of linear systems can also be represented using graphical structures and the corresponding theorems of the alternative can then be stated in terms of properties of the graphical structure. For instance, it is well-known that a system of difference constraints DCS can be represented as a constraint network such that the DCS is feasible if and only if there does not exist a negative cost cycle in the network. In this paper, we provide a new graphical constraint network representation of Unit Two Variable Per Inequality UTVPI constraints. This constraint network representation permits us to derive a theorem of the alternative for the feasibility of UTVPI systems. UTVPI constraints find applications in a number of domains, including but not limited to program verification, abstract interpretation, and array bounds checking. Theorems of the Alternative find primary use in the design of certificates in certifying algorithms. It follows that our work is important from this perspective.