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Complexity Analyses of Bienstock–Zuckerberg and Lasserre Relaxations on the Matching and Stable Set Polytopes

  • Conference paper
Integer Programming and Combinatoral Optimization (IPCO 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6655))

Abstract

Many hierarchies of lift-and-project relaxations for 0,1 integer programs have been proposed, two of the most recent and strongest being those by Lasserre in 2001, and Bienstock and Zuckerberg in 2004. We prove that, on the LP relaxation of the matching polytope of the complete graph on (2n + 1) vertices defined by the nonnegativity and degree constraints, the Bienstock–Zuckerberg operator (even with positive semidefiniteness constraints) requires \(\Theta(\sqrt{n})\) rounds to reach the integral polytope, while the Lasserre operator requires Θ(n) rounds. We also prove that Bienstock–Zuckerberg operator, without the positive semidefiniteness constraint requires approximately n/2 rounds to reach the stable set polytope of the n-clique, if we start with the fractional stable set polytope. As a by-product of our work, we consider a significantly strengthened version of Sherali–Adams operator and a strengthened version of Bienstock–Zuckerberg operator. Most of our results also apply to these stronger operators.

This research was supported in part by an OGSST Scholarship, a Sinclair Scholarship, an NSERC Scholarship and NSERC Discovery Grants.

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Yu Hin Au & Levent Tunçel

Authors
  1. Yu Hin Au
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  2. Levent Tunçel
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Editors and Affiliations

  1. Mathematical Sciences Department, IBM T. J. Watson Research Center, 10598, Yorktown Heights, NY, USA

    Oktay Günlük

  2. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600, Eindhoven, MB, The Netherlands

    Gerhard J. Woeginger

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Au, Y.H., Tunçel, L. (2011). Complexity Analyses of Bienstock–Zuckerberg and Lasserre Relaxations on the Matching and Stable Set Polytopes. In: Günlük, O., Woeginger, G.J. (eds) Integer Programming and Combinatoral Optimization. IPCO 2011. Lecture Notes in Computer Science, vol 6655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20807-2_2

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  • DOI: https://doi.org/10.1007/978-3-642-20807-2_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20806-5

  • Online ISBN: 978-3-642-20807-2

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