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, ) and Aaronson (SIAM J. Comput. 35(4):804-824, ) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804-824, ).
All Science Journal Classification (ASJC) codes
- Computer Science(all)
- Computer Science Applications
- Applied Mathematics
- Local search
- Quantum computing
- Query complexity