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Restricted Cutting Plane Proofs in Horn Constraint Systems

  • Hans Kleine Büning
  • Piotr WojciechowskiEmail author
  • R. Chandrasekaran
  • K. Subramani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11715)

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.

Notes

Acknowledgments

This work was supported by the Air Force Research Laboratory under US Air Force contract FA8750-16-3-6003. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

This research was made possible by NASA WV EPSCoR Grant # NNX15AK74A and by the AFOSR through grant FA9550-19-1-017.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Hans Kleine Büning
    • 1
  • Piotr Wojciechowski
    • 2
    Email author
  • R. Chandrasekaran
    • 3
  • K. Subramani
    • 2
  1. 1.Computer Science InstituteUniversity of PaderbornPaderbornGermany
  2. 2.LCSEEWest Virginia UniversityMorgantownUSA
  3. 3.Computer Science and EngineeringThe University of Texas at DallasRichardsonUSA

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