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[4] Ald and Aaronson [1] Aar and solves the main open problem in Aar.
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机译：假设 f 是有限集 V 上的整数函数。我们称 f 的无向图 G（V，E） 邻域结构。查找 f 的局部最小值的问题可以表述为：对于固定邻域结构 G（V，E），找到顶点 x ∈ V ，使得 f（x）不大于 f 对 x 的某个邻居取的任何值。 。该算法的复杂性通过以下形式的问题数量来衡量：“ x 上的 f 的值是多少？”我们表明问题的确定性，随机性和量子查询复杂性与多项式相关。这概括了Aldous [4] Ald和Aaronson [1] Aar的早期结果，并解决了Aar中的主要开放问题。
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