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

Abstract

In 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.

This research was supported in part by the Air-Force Office of Scientific Research through Grant FA9550-19-1-0177 and in part by the Air-Force Research Laboratory, Rome through Contract FA8750-17-S-7007.