Quantum and classical query complexities of local search are polynomially related

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 [?] and Aaronson [?] and solves the main open problem in [?].

• Type: Conference
• Year: 2004
• Where: STOC
• Authors: Miklos Santha, Mario Szegedy
• Keywords: Algorithms, Fixed Neighborhood Structure, Integer Valued Function, Main Open Problem, STOC 2004