Discrete Applied Mathematics
Local expansion of vertex-transitive graphs and random generation in finite groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Quantum lower bounds by quantum arguments
Journal of Computer and System Sciences - Special issue on STOC 2000
Lower Bounds for Local Search by Quantum Arguments
SIAM Journal on Computing
New upper and lower bounds for randomized and quantum local search
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Quantum and Classical Query Complexities of Local Search Are Polynomially Related
Algorithmica - Special Issue: Quantum Computation; Guest Editors: Frédéric Magniez and Ashwin Nayak
On the Quantum Query Complexity of Local Search in Two and Three Dimensions
Algorithmica - Special Issue: Quantum Computation; Guest Editors: Frédéric Magniez and Ashwin Nayak
Enhanced algorithms for Local Search
Information Processing Letters
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We study the problem of local search on a graph. Given a real-valued black-box functionf on the graph's vertices, this is the problem of determining a local minimum of f--a vertex v for which f(v) is no more than f evaluated at any of v's neighbors. In1983, Aldous gave the first strong lower bounds for the problem, showing that anyrandomized algorithm requires Ω(2n/2-o(n)) queries to determine a local minimum onthe n-dimensional hypercube. The next major step forward was not until 2004 whenAaronson, introducing a new method for query complexity bounds, both strengthened thislower bound to Ω(2n/2/n2) and gave an analogous lower bound on the quantum querycomplexity. While these bounds are very strong, they are known only for narrow familiesof graphs (hypercubes and grids). We show how to generalize Aaronson's techniques inorder to give randomized (and quantum) lower bounds on the query complexity of localsearch for the family of vertex-transitive graphs. In particular, we show that for anyvertex-transitive graph G of N vertices and diameter d, the randomized and quantumquery complexities for local search on G are Ω (√N/dlogN) and (4√N / √dlogN),respectively.