By Andrej Bogdanov, Luca Trevisan

Average-Case Complexity is a radical survey of the average-case complexity of difficulties in NP. The learn of the average-case complexity of intractable difficulties started within the Nineteen Seventies, inspired by way of exact purposes: the advancements of the principles of cryptography and the hunt for ways to "cope" with the intractability of NP-hard difficulties. This survey appears at either, and usually examines the present country of data on average-case complexity. Average-Case Complexity is meant for students and graduate scholars within the box of theoretical computing device technological know-how. The reader also will find a variety of effects, insights, and facts innovations whose usefulness is going past the learn of average-case complexity.

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Since the reduction needs to access witnesses for membership of its queries, we formalize it as a reduction between search problems. We only consider the case when one is reducing to a problem with respect to the uniform distribution, as this is our case of interest. For two distributional problems (L, D) and (L , U) in (NP, PSamp), a randomized heuristic search reduction from (L, D) to (L , U) is an algorithm R that takes an input x and a parameter n and runs in time polynomial in n, such that for every n and every x, there exists a set Vx ⊆ Supp R(x; n) (corresponding to the “uniquely decodable” queries) with the following properties: (1) Disjointness: There is a polynomial p such that for every n, Vx ⊆ {0, 1}p(n) and the sets Vx are pairwise disjoint.

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.

2. Reducing search to decision 47 distribution, does every search problem in NP also have efficient algorithms with respect to the uniform distribution? We show a result of Ben-David et al. that establishes the equivalence of search and decision algorithms for NP with the uniform distribution. 1. Let us recall the common argument used to establish the equivalence of NP-hardness for search and decision problems in the worst-case setting, and see why this argument fails to carry over directly to the average-case setting.