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No Small Linear Program Approximates Vertex Cover Within a Factor 2 − ɛ

Author

Listed:
  • Abbas Bazzi

    (School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, Lausanne, France CH-1015)

  • Samuel Fiorini

    (Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium B-1050)

  • Sebastian Pokutta

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30318)

  • Ola Svensson

    (School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, Lausanne, France CH-1015)

Abstract

The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev [Khot S, Regev O (2008) Vertex cover might be hard to approximate to within 2 − ɛ . J. Comput. System Sci. 74(3):335–349] proved that the problem is NP-hard to approximate within a factor 2 − ɛ , assuming the unique games conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best inapproximability result for the problem is due to Dinur and Safra [Dinur I, Safra S (2005) On the hardness of approximating minimum vertex cover. Ann. Math. 162(1):439–485]: vertex cover is NP-hard to approximate within a factor 1.3606. We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates the vertex cover within a factor 2 − ɛ has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as semidefinite programming relaxations) that approximate the independent set problem within any constant factor have a super-polynomial size.

Suggested Citation

  • Abbas Bazzi & Samuel Fiorini & Sebastian Pokutta & Ola Svensson, 2019. "No Small Linear Program Approximates Vertex Cover Within a Factor 2 − ɛ," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 147-172, February.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:1:p:147-172
    DOI: 10.1287/moor.2017.0918
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    References listed on IDEAS

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    1. Gábor Braun & Samuel Fiorini & Sebastian Pokutta & David Steurer, 2015. "Approximation Limits of Linear Programs (Beyond Hierarchies)," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 756-772, March.
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    Cited by:

    1. Vasily V. Gusev, 2021. "Set-weighted games and their application to the cover problem," HSE Working papers WP BRP 247/EC/2021, National Research University Higher School of Economics.
    2. Gusev, Vasily V., 2023. "Set-weighted games and their application to the cover problem," European Journal of Operational Research, Elsevier, vol. 305(1), pages 438-450.

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