### 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]).

Original language | English (US) |
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Pages (from-to) | 557-575 |

Number of pages | 19 |

Journal | Algorithmica (New York) |

Volume | 55 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1 2009 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Keywords

- Local search
- Quantum computing
- Query complexity