A Polynomial Time Algorithm for Read-Once Certification of Linear Infeasibility in UTVPI Constraints

Abstract

In this paper, we discuss the design and analysis of polynomial time algorithms for two problems associated with a linearly infeasible system of Unit Two Variable Per Inequality (UTVPI) constraints, viz., (a) the read-once refutation (ROR) problem, and (b) the literal-once refutation (LOR) 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: \(\mathbf{A \cdot x \le b}\). These constraints 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 \(\mathbf{y}=[y_{1},y_{2},\ldots , y_{m}]^{T} \in \mathfrak {R}^{m}_{+}\), 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 transformation results in an algorithm that runs in in time polynomial in the size of the input UCS. Additionally, we design a polynomial time algorithm (also via a reduction to the MWPM problem) for a variant of read-once resolution called literal-once resolution. The advantage of reducing our problems to the MWPM problem is that we can leverage recent advances in algorithm design for the MWPM problem towards solving the ROR and LOR problems in UCSs. Finally, we show that another variant of read-once refutation problem called the non-literal read-once refutation (NLROR) problem is NP-complete in UCSs.