Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
Discrete Applied Mathematics
On total functions, existence theorems and computational complexity
Theoretical Computer Science
On the deterministic complexity of searching local maxima
Discrete Applied Mathematics - Special issue: local optimization
Dividing and conquering the square
Discrete Applied Mathematics - Special issue: local optimization
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Quantum lower bounds by quantum arguments
Journal of Computer and System Sciences - Special issue on STOC 2000
Polynomial Degree vs. Quantum Query Complexity
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
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
Lower Bounds for Randomized and Quantum Query Complexity Using Kolmogorov Arguments
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Approximate Local Search in Combinatorial Optimization
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
Enhanced algorithms for local search
Information Processing Letters
On the power of Ambainis lower bounds
Theoretical Computer Science
All quantum adversary methods are equivalent
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
New upper and lower bounds for randomized and quantum local search
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Quantum Separation of Local Search and Fixed Point Computation
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Quantum and Randomized Lower Bounds for Local Search on Vertex-Transitive Graphs
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Quantum and randomized lower bounds for local search on vertex-transitive graphs
Quantum Information & Computation
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Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to combinatorial optimization, complexity theory and many other areas in theoretical computer science. In this paper, we study the problem in the randomized and quantum query models and give new lower and upper bound techniques in both models.The lower bound technique works for any graph that contains a product graph as a subgraph. Applying it to the Boolean hypercube (0, 1)n and the constant dimensional grids [n]d, two particular product graphs that recently drew much attention, we get the following tight results: RLS((0, 1)n) = Θ(2n/2n1/2), QLS((0, 1)n) = Θ(2n/3n1/6); RLS([n]d) = Θ(nd/2), ∀ d ≥ 4, QLS([n]d) = Θ(nd/3), ∀ d ≥ 6. Here RLS(G) and QLS(G) are the randomized and quantum query complexities of Local Search on G, respectively. These improve the previous results by Aaronson [2], Ambainis (unpublished) and Santha and Szegedy [20].Our new algorithms work well when the underlying graph expands slowly. As an application to [n]2, a new quantum algorithm using O(☂n(log log n)1.5) queries is given. This improves the previous best known upper bound of O(n2/3) (Aaronson, [2]), and implies that Local Search on grids exhibits different properties in low dimensions.