By Andrej Bogdanov, Luca Trevisan
Average-Case Complexity is a radical survey of the average-case complexity of difficulties in NP. The research of the average-case complexity of intractable difficulties all started within the Nineteen Seventies, inspired by way of unique functions: the advancements of the rules of cryptography and the quest for tactics to "cope" with the intractability of NP-hard difficulties. This survey seems at either, and usually examines the present country of information on average-case complexity. Average-Case Complexity is meant for students and graduate scholars within the box of theoretical machine technology. The reader also will find a variety of effects, insights, and facts thoughts whose usefulness is going past the examine of average-case complexity.
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Average-Case Complexity is an intensive survey of the average-case complexity of difficulties in NP. The learn of the average-case complexity of intractable difficulties started within the Seventies, prompted by means of distinctive purposes: the advancements of the principles of cryptography and the hunt for ways to "cope" with the intractability of NP-hard difficulties.
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Additional resources for Average-case complexity
To rule out the existence of general one-way functions in this setting, it is sufficient (by H˚ astad et al. ) to show that no pseudorandom generators exist. We argue that this is the case in the errorless setting, that is under the assumption (NP, U) ⊆ AvgBPP. ” An errorless algorithm A for this problem must always answer “yes” or ⊥ when the input is chosen according to Gn (Un−1 ). On the other hand, A(y; n, 1/4) must answer “no” on at least a 14 fraction of inputs y ∼ Un , since at least a 12 fraction of such inputs is outside the image of Gn , and the algorithm is allowed to fail on no more than a 14 fraction of inputs.
2 Heuristic algorithms versus heuristic schemes When defining average-case complexity classes, we distinguished between heuristic algorithms and heuristic schemes: For heuristic algorithms, we fix a failure probability δ and require that the algorithm succeeds on all but a δ-fraction of the instances. For heuristic schemes, we require a single algorithm that works for all δ, but we allow the running time to grow as a function of 1/δ. It is clear that if a distributional problem has a heuristic scheme, then it has heuristic algorithms with failure probability δ(n) = n−c for every c > 0.
Given a decision oracle for NP, and an instance x of an NP-language L, a search algorithm for x finds a witness by doing binary search for the lexicographically smallest w such that the oracle answers “yes” on the NP-query: (x, w): Is there an L-witness for x that is lexicographically at most w? To see why this reduction is useless in the average-case setting with respect to the uniform distribution, fix the lexicographically smallest witness wx for every x ∈ L, and suppose that the average-case decision oracle answers all queries correctly, except those (x, w) where the distance between w and wx in the lexicographic order is small.
Average-case complexity by Andrej Bogdanov, Luca Trevisan