Farkas Bounds on Horn Constraint Systems

Abstract

In this paper, we analyze the copy complexity of unsatisfiable Horn constraint systems, under the ADD refutation system. Recall that a linear constraint of the form \(\sum _{i=1}^{n} a_{i}\cdot x_{i} \ge b\), is said to be a horn constraint if all the \(a_{i} \in \{0,1,-1\}\) and at most one of the \(a_{i}\)s is positive. A conjunction of such constraints is called a Horn constraint system (HCS). Horn constraints arise in a number of domains including, but not limited to, program verification, power systems, econometrics, and operations research. The ADD refutation system is both sound and complete. Additionally, it is the simplest and most natural refutation system for refuting the feasibility of a system of linear constraints. The copy complexity of an infeasible linear constraint system (not necessarily Horn) in a refutation system, is the minimum number of times each constraint needs to be replicated, in order to obtain a read-once refutation. We show that for an HCS with n variables and m constraints, the copy complexity is at most \(2^{n-1}\), in the ADD refutation system. Additionally, we analyze bounded-width HCSs from the perspective of copy complexity. Finally, we provide an empirical analysis of an integer programming formulation of the copy complexity problem in HCSs. (An extended abstract was published in FroCos 2021 [26].)