Exact and Parameterized Algorithms for Read-Once Refutations in Horn Constraint Systems

Abstract

AbstractIn this paper, we discuss exact and parameterized algorithms for the problem of finding a read-once refutation in an unsatisfiable Horn Constraint System (HCS). Recall that a linear constraint system \(\mathbf{A \cdot x \ge b}\) is said to be a Horn constraint system, if each entry in \(\mathbf{A}\) belongs to the set \(\{0,1,-1\}\) and at most one entry in each row of \(\mathbf{A}\) is positive. In this paper, we examine the importance of constraints in which more variables have negative coefficients than have positive coefficients. There exist several algorithms for checking whether a Horn constraint system is feasible. To the best of our knowledge, these algorithms are not certifying, i.e., they do not provide a certificate of infeasibility. Our work is concerned with providing a specialized class of certificates called “read-once refutations”. In a read-once refutation, each constraint defining the HCS may be used at most once in the derivation of a refutation. The problem of checking if an HCS has a read-once refutation (HCS ROR) has been shown to be NP-hard. We analyze the HCS ROR problem from two different algorithmic perspectives, viz., parameterized algorithms and exact exponential algorithms.