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 started within the Nineteen Seventies, influenced via targeted functions: the advancements of the rules of cryptography and the hunt for tactics to "cope" with the intractability of NP-hard difficulties. This survey seems 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 machine technological know-how. The reader also will find a variety of effects, insights, and evidence suggestions whose usefulness is going past the examine of average-case complexity.

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1. (reduction between distributional problems) Let (L, D) and (L , D ) be two distributional problems. We say that (L, D) reduces to (L , D ), and write (L, D) ≤AvgP (L , D ), if there is a function f that for every n, on input x in the support of Dn and parameter n, can be computed in time polynomial in n and 35 36 A Complete Problem for Computable Ensembles (1) (Correctness) x ∈ L if and only if f (x; n) ∈ L . (2) (Domination) There are polynomials p and m such that, for every n and every y in the support of Dm(n) , Dn (x) ≤ p(n)Dm(n) (y).

If, on the other hand, Dn (x) > 2−|x| , let y be the string that precedes x in lexicographic order among the strings in {0, 1}n and let p = fDn (y) (if x is the empty string, then we let p = 0). Then we define C(x; n) = 1z. Here z is the longest common prefix of fDn (x) and p when both are written out in binary. Since fDn is computable in polynomial time, so is z. C is injective because only two binary strings s1 and s2 can have the same longest common prefix z; a third string s3 sharing z as a prefix must have a longer prefix with either s1 or s2 .

For technical reasons, our algorithms are given, in addition to the input x, a parameter n corresponding to the distribution Dn from which x was sampled. We write A(x; n) to denote the output of algorithm A on input x and parameter n. 2. Heuristic and errorless algorithms 21 for every n, where tA (x; n) is the running time of A on input x and parameter n. Such a definition is problematic because there are algorithms that we would intuitively consider to be “typically efficient” but whose expected running time is superpolynomial.

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