Restricted Cutting Plane Proofs in Horn Constraint Systems

Abstract

In this paper, we investigate variants of cutting plane proof systems for a class of integer programs called Horn constraint systems (HCS). Briefly a system of linear inequalities \(\mathbf{A \cdot x \ge b}\) is called a Horn constraint system, if each entry in \(\mathbf{A}\) belongs to the set \(\{0,1,-1\}\) and furthermore there is at most one positive entry per row. Our focus is on deriving refutations i.e., proofs of unsatisfiability of such programs in variants of the cutting plane proof system. Horn systems generalize Horn formulas, i.e., CNF formulas with at most one positive literal per clause. A Horn system which results from rewriting a Horn clausal formula is called a Horn clausal constraint system (HClCS). The cutting plane calculus (CP) is a well-known calculus for deciding the unsatisfiability of propositional CNF formulas and integer programs. Usually, CP consists of the addition rule (ADD) and the division rule (DIV). We show that the cutting plane calculus with the addition rule only (CP-ADD) does not require constraints of the form \(0 \le x_i \le 1\). We also investigate the existence of read-once refutations in Horn clausal constraint systems in the cutting plane proof system. We show that read-once refutations are incomplete and furthermore the problem of checking for the existence of a read-once refutation in an arbitrary Horn clausal system is NP-complete.