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On the Black-Box Complexity of Sperner’s Lemma

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Abstract

We present several results on the complexity of various forms of Sperner’s Lemma in the black-box model of computing. We give a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an \(O(\sqrt{n})\) deterministic query algorithm for the black-box version of the problem 2D-SPERNER, a well studied member of Papadimitriou’s complexity class PPAD. This upper bound matches the \(\Omega(\sqrt{n})\) deterministic lower bound of Crescenzi and Silvestri. The tightness of this bound was not known before. In another result we prove for the same problem an \(\Omega(\sqrt[4]{n})\) lower bound for its probabilistic, and an \(\Omega(\sqrt[8]{n})\) lower bound for its quantum query complexity, showing that all these measures are polynomially related.

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Authors and Affiliations

  1. Budapest University of Technology and Economics, P.O. Box 91, 1521, Budapest, Hungary

    Katalin Friedl

  2. Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, 1518, Budapest, Hungary

    Gábor Ivanyos

  3. CNRS–LRI, Université Paris-Sud, 91405, Orsay, France

    Miklos Santha

  4. Centre for Quantum Technologies, National University of Singapore, Singapore, 117543, Singapore

    Miklos Santha

  5. LRI, Université Paris-Sud, 91405, Orsay, France

    Yves F. Verhoeven

  6. ENST, 46 rue Barrault, 75013, Paris, France

    Yves F. Verhoeven

Authors
  1. Katalin Friedl
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  2. Gábor Ivanyos
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  3. Miklos Santha
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  4. Yves F. Verhoeven
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Corresponding author

Correspondence to Miklos Santha.

Additional information

Research supported by the European Commission IST Integrated Project Qubit Application (QAP) 015848, the OTKA grants T42559 and T46234, and by the ANR Blanc AlgoQP grant of the French Research Ministry.

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Friedl, K., Ivanyos, G., Santha, M. et al. On the Black-Box Complexity of Sperner’s Lemma. Theory Comput Syst 45, 629–646 (2009). https://doi.org/10.1007/s00224-008-9121-2

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  • DOI: https://doi.org/10.1007/s00224-008-9121-2

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