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High Degree Sum of Squares Proofs, Bienstock-Zuckerberg Hierarchy and CG Cuts

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Integer Programming and Combinatorial Optimization (IPCO 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10328))

Abstract

Chvátal-Gomory (CG) cuts captures useful and efficient linear programs that the bounded degree Lasserre/Sum-of-Squares (\({\textsc {sos}}\)) hierarchy fails to capture. We present an augmented version of the \({\textsc {sos}}\) hierarchy for 0/1 integer problems that implies the Bienstock-Zuckerberg hierarchy by using high degree polynomials (when expressed in the standard monomial basis). It follows that for a class of polytopes (e.g. set covering and packing problems), the \({\textsc {sos}}\) approach can optimize, up to an arbitrarily small error, over the polytope resulting from any constant rounds of CG cuts in polynomial time.

Supported by the Swiss National Science Foundation project 200020-169022 “Lift and Project Methods for Machine Scheduling Through Theory and Experiments”.

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Notes

  1. 1.

    In some research communities such linear functional is called pseudo-expectation.

  2. 2.

    This is the standard bounded degree \({\textsc {sos}}\) proof system.

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Authors and Affiliations

  1. IDSIA, 6928, Manno, Switzerland

    Monaldo Mastrolilli

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  1. Monaldo Mastrolilli
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Correspondence to Monaldo Mastrolilli .

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Editors and Affiliations

  1. Ecole Polytechnique Federale de Lausanne - EPFL, Lausanne, Switzerland

    Friedrich Eisenbrand

  2. University of Waterloo, Waterloo, Ontario, Canada

    Jochen Koenemann

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Mastrolilli, M. (2017). High Degree Sum of Squares Proofs, Bienstock-Zuckerberg Hierarchy and CG Cuts. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_33

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  • DOI: https://doi.org/10.1007/978-3-319-59250-3_33

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59249-7

  • Online ISBN: 978-3-319-59250-3

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