A separator theorem for graphs of bounded genus
Journal of Algorithms
Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
Discrete Applied Mathematics
On the complexity of local search
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Dividing and conquering the square
Discrete Applied Mathematics - Special issue: local optimization
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Lower bounds for local search by quantum arguments
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Quantum and classical query complexities of local search are polynomially related
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
New upper and lower bounds for randomized and quantum local search
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
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Let G = (V, E) be a finite graph, and f : V → N be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity O(log n) ċ d + O(√g) ċ √n, so that we obtain a deterministic query complexity of d + O(√g) ċ √n where n is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of O(√d) + O(4√g) ċ 4√nlog logn. Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs.