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On the lengths of tree-like and Dag-like cutting plane refutations of Horn constraint systems

Horn constraint systems and cutting plane refutations

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Abstract

In this paper, we investigate the properties of cutting plane based refutations for a class of integer programs called Horn constraint systems (HCSs). Briefly, a system of linear inequalities A ⋅x ≥b is called a Horn constraint system, if each entry in 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 using cutting planes as a proof system. We also look at several properties of these refutations. HCSs can be considered a more general form of Horn formulas, i.e., CNF formulas with at most one positive literal per clause. Cutting plane calculus (CP) is a well-known calculus for deciding the unsatisfiability of propositional CNF formulas and integer programs. Usually, CP consists of a pair of inference rules. These are called the addition rule (ADD) and the division rule (DIV). In this paper, we show that cutting plane calculus is still complete for HCSs when every intermediate constraint is required to be Horn. We also investigate the lengths of cutting plane proofs for Horn constraint systems.

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Acknowledgments

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

We would like to thank Hans Kleine Büning for his insights into the problems examined in this paper.

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  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    P. Wojciechowski & K. Subramani

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  1. P. Wojciechowski
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  2. K. Subramani
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Correspondence to K. Subramani.

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An extended abstract of this work was presented at FSTTCS 2019 [24].

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Wojciechowski, P., Subramani, K. On the lengths of tree-like and Dag-like cutting plane refutations of Horn constraint systems. Ann Math Artif Intell 90, 979–998 (2022). https://doi.org/10.1007/s10472-022-09800-7

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