Read-once refutations in Horn constraint systems: an algorithmic approach

Abstract

Abstract In this paper, we discuss exact and parameterized algorithms for the problem of finding a read-once refutation (ROR) in an unsatisfiable Horn constraint system (HCS). Recall that a linear constraint system $\mathbf {A \cdot x \ge b}$ is said to be an HCS if each entry in $\textbf {A}$ belongs to the set $\{0,1,-1\}$ and at most one entry in each row of $\textbf {A}$ is positive. In this paper, we examine the importance of constraints in which more variables have negative coefficients than positive coefficients. In particular, we study the impact of the proportion of these ‘net-negative’ constraints has on the difficulty of finding RORs. There exist several algorithms for checking whether an HCS 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 an ROR, 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 an ROR has been shown to be NP-hard. We analyse the HCS ROR problem from three different algorithmic perspectives, viz., parameterized algorithms, exact exponential algorithms and approximation algorithms. In particular, we show that the HCS ROR problem is fixed-parameter tractable (FPT) with respect to the number of constraints in the system that have more variables with negative coefficient than variables with positive coefficient. Additionally, we show that the HCS ROR problem becomes easy when this parameter is both small and large. We also derive an algorithm that runs in time $O(1.66^{m})$, where $m$ is the number of constraints in the HCS. On the lower-bound side, we derive a lower bound on the algorithmic resources needed for this problem using the Exponential Time Hypothesis. We also establish that the HCS ROR problem does not have a polynomial kernel when the number of constraints with three or more variables in a refutation is used as a parameter. Finally, we show that the problem of approximating the length of the shortest ROR in an HCS is NPO PB-complete1.