A Graphical Analysis of Integer Infeasibility in UTVPI Constraints

Abstract

In this paper, we discuss a theorem of the alternative for integer feasibility in a class of constraints called Unit Two Variable Per Inequality (UTVPI) constraints. In general, a theorem of the alternative gives two systems of constraints such that exactly one system is feasible. Theorems of the alternative for linear feasibility have been discussed extensively in the literature. If a theorem of the alternative provides a “succinct” certificate of infeasibility, it is said to be compact. In general, theorems of the alternative for linear feasibility are compact (see Farkas’ lemma for instance). However, compact theorems of the alternative cannot exist for integer feasibility in linear programs unless NP$\,=\,$coNP. A second feature of a theorem of the alternative is its form. Typically, theorems of the alternative connect pairs of linear systems. A graphical theorem of the alternative, on the other hand, connects infeasibility in a linear system to the existence of particular paths in an appropriately constructed constraint network. Graphical theorems of the alternative are known to exist for selected classes of linear programs. In this paper, we detail a compact, graphical theorem of the alternative for integer feasibility in UTVPI constraints.