Read-Once Certification of Linear Infeasibility in UTVPI Constraints

Abstract

In this paper, we discuss the design and analysis of a polynomial time algorithm for a problem associated with a linearly infeasible system of Unit Two Variable Per Inequality (UTVPI) constraints, viz., the read-once refutation (ROR) problem. Recall that a UTVPI constraint is a linear relationship of the form: \(a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{ij}\), where \(a_{i},a_{j} \in \{0,1,-1\}\). A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: \(\mathbf{A \cdot x \le c}\). These constraint find applications in a host of domains including but not limited to operations research and program verification. For the linear system \(\mathbf{A\cdot x \le b}\), a refutation is a collection of m variables \(\mathbf{y}=[y_{1},y_{2},\ldots , y_{m}]^{T} \in \mathbb {R}^{m}_{+}\), such that \(\mathbf{y\cdot A =0}\), \(\mathbf{y \cdot b } < 0\). Such a refutation is guaranteed to exist for any infeasible linear program, as per Farkas’ lemma. The refutation is said to be read-once, if each \(y_{i} \in \{0,1\}\). Read-once refutations are incomplete in that their existence is not guaranteed for infeasible linear programs, in general. Indeed they are not complete, even for UCSs. Hence, the question of whether an arbitrary UCS has an ROR is both interesting and non-trivial. In this paper, we reduce this problem to the problem of computing a minimum weight perfect matching (MWPM) in an undirected graph. This results in an algorithm that runs in time polynomial in the size of the input UCS.