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On the Width of Semialgebraic Proofs and Algorithms

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  • Alexander Razborov

    (Departments of Mathematics and Computer Science, University of Chicago, Chicago, Illinois 60637; and Steklov Mathematical Institute, Moscow, Russia 117418)

Abstract

In this paper we study width of semialgebraic proof systems and various cut-based procedures in integer programming. We focus on two important systems: Gomory-Chvátal cutting planes and Lovász-Schrijver lift-and-project procedures. We develop general methods for proving width lower bounds and apply them to random k-CNFs and several popular combinatorial principles, like the perfect matching principle and Tseitin tautologies. We also show how to apply our methods to various combinatorial optimization problems. We establish a “supercritical” trade-off between width and rank, that is we give an example in which small width proofs are possible but require exponentially many rounds to perform them.

Suggested Citation

  • Alexander Razborov, 2017. "On the Width of Semialgebraic Proofs and Algorithms," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1106-1134, November.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:4:p:1106-1134
    DOI: 10.1287/moor.2016.0840
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    References listed on IDEAS

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    1. Sanjeeb Dash, 2005. "Exponential Lower Bounds on the Lengths of Some Classes of Branch-and-Cut Proofs," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 678-700, August.
    2. Schrijver, A, 1980. "On Cutting Planes," University of Amsterdam, Actuarial Science and Econometrics Archive 293054, University of Amsterdam, Faculty of Economics and Business.
    3. Michel X. Goemans & Levent Tunçel, 2001. "When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures?," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 796-815, November.
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