On the Parallel Complexity of Constrained Read-Once Refutations in UTVPI Constraint Systems

Abstract

This paper is concerned with the parallel complexity of the constraint-required read-once refutation (CROR) problem in Unit Two Variable Per Inequality (UTVPI) constraint systems. Recall that a UTVPI constraint is a linear inequality of the form: $$a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{k}$$ , where $$a_{i},a_{j} \in \{0,1,-1\}$$ and $$b_{k} \in \mathbb {Z}$$ . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: $$\mathbf{A \cdot x \le b}$$ . UTVPI constraints are used in many domains including operations research and program verification. A refutation is a proof of infeasibility. A read-once refutation (ROR) is a refutation in which each constraint is used at most once. We focus on a variant of the ROR problem in which we specify which constraints a refutation is required to use. This variant is known as the CROR problem. In this paper, we provide NC reductions between the CROR problem in UCSs and the decision version of the minimum weight perfect matching problem.