## Average-Case Complexity (Foundations and Trends(r) in by Andrej Bogdanov, Luca Trevisan

By Andrej Bogdanov, Luca Trevisan

Average-Case Complexity is an intensive survey of the average-case complexity of difficulties in NP. The examine of the average-case complexity of intractable difficulties begun within the Seventies, stimulated through distinctive functions: the advancements of the rules of cryptography and the quest for ways to "cope" with the intractability of NP-hard difficulties. This survey appears at either, and usually examines the present nation of information on average-case complexity. Average-Case Complexity is meant for students and graduate scholars within the box of theoretical laptop technology. The reader also will find a variety of effects, insights, and facts innovations whose usefulness is going past the examine of average-case complexity.

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For every i = 1, . . , m(|x|), where m(n) is the length of a witness on inputs of length n. ) Given 48 Decision Versus Search and One-Way Functions a worst-case decision oracle for this NP language, the sequence of oracle answers on input x ∈ L allows the search algorithm to recover all the bits of the unique witness w. In this setting, the reduction also works well on average: Given an average-case decision oracle that works on a 1 − δ/m(n) fraction of inputs (x, i), where |x| = n and i ≤ m(n), the search algorithm is able to recover witnesses (if they exist) on a 1 − δ fraction of inputs x ∼ Un .

Randomized heuristic classes) We say that (L, D) is in Heurδ BPTIME(t(n)) if there is a randomized errorless algorithm A of failure probability at most δ(n) and of running time at most t(n) on inputs in the support of Dn . We define Heurδ BPP and HeurBPP in the obvious way. For all classes of the type Avgδ C and Heurδ C defined above, we define Avgneg C and Heurneg C as their union over all negligible functions δ, respectively. 30 Definitions of “Efficient on Average” For the non-uniform and randomized heuristic classes, we have the standard containments AvgC ⊆ HeurC.

The value K(x) is called the (prefix-free) Kolmogorov complexity of x. The universal probability distribution K is defined so that the probability of a string x is 2−K(x) . Observe that x 2−K(x) ≤ 1 since the representation of (M, w) is prefix free. ) Finally, let {Kn } be the ensemble of distributions, where Kn is the distribution K conditioned on strings of length n. It turns out that for every language L, solving L well on average with a heuristic algorithm is as hard as solving L well on the worst case.