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0/1 Polytopes with Quadratic Chvátal Rank

Author

Listed:
  • Thomas Rothvoß

    (University of Washington, Seattle, 98105)

  • Laura Sanità

    (University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada)

Abstract

For a polytope P , the Chvátal closure P ′ ⊆ P is obtained by simultaneously strengthening all feasible inequalities cx ⩽ β (with integral c ) to cx ⩽ ⌊ β ⌋. The number of iterations of this procedure that are needed until the integral hull of P is reached is called the Chvátal rank. If P ⊆ [0, 1] n , then it is known that O ( n 2 log n ) iterations always suffice and at least (1 + 1/ e − o (1)) n iterations are sometimes needed, leaving a huge gap between lower and upper bounds.We prove that there is a polytope contained in the 0/1 cube that has Chvátal rank Ω( n 2 ), closing the gap up to a logarithmic factor. In fact, even a superlinear lower bound was mentioned as an open problem by several authors. Our choice of P is the convex hull of a semi-random Knapsack polytope and a single fractional vertex. The main technical ingredient is linking the Chvátal rank to simultaneous Diophantine approximations w.r.t. the ‖·‖ 1 -norm of the normal vector defining P .

Suggested Citation

  • Thomas Rothvoß & Laura Sanità, 2017. "0/1 Polytopes with Quadratic Chvátal Rank," Operations Research, INFORMS, vol. 65(1), pages 212-220, February.
  • Handle: RePEc:inm:oropre:v:65:y:2017:i:1:p:212-220
    DOI: 10.1287/opre.2016.1549
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    References listed on IDEAS

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    1. Schrijver, A, 1980. "On Cutting Planes," University of Amsterdam, Actuarial Science and Econometrics Archive 293054, University of Amsterdam, Faculty of Economics and Business.
    2. Monique Laurent, 2003. "A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 470-496, August.
    3. Daniel Dadush & Santanu S. Dey & Juan Pablo Vielma, 2011. "The Chvátal-Gomory Closure of a Strictly Convex Body," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 227-239, May.
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    Keywords

    integer programming; Chvátal-gomory cuts;

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