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Quantum and Classical Query Complexities of Local Search Are Polynomially Related

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Abstract

Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x∈V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x?” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous (Ann. Probab. 11(2):403–413, [1983]) and Aaronson (SIAM J. Comput. 35(4):804–824, [2006]) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804–824, [2006]).

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Authors and Affiliations

  1. LRI, Université Paris-Sud, CNRS, 91405, Orsay, France

    Miklos Santha

  2. Rutgers University, Piscataway, NJ, 08854-8019, USA

    Mario Szegedy

Authors
  1. Miklos Santha
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  2. Mario Szegedy
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Corresponding author

Correspondence to Miklos Santha.

Additional information

Research of M. Santha was supported by the European Commission IST projects RESQ 37559 and QAP 015848, and by the ANR Blanc AlgoQP grant of the French Research Ministry.

Research of M. Szegedy was supported by NSF grant 0105692 and the European Commission IST project RESQ 37559. The research was done while the author was visiting LRI, Université Paris–Sud, CNRS.

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Santha, M., Szegedy, M. Quantum and Classical Query Complexities of Local Search Are Polynomially Related. Algorithmica 55, 557–575 (2009). https://doi.org/10.1007/s00453-008-9169-z

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  • DOI: https://doi.org/10.1007/s00453-008-9169-z

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