Average-case complexity by Andrej Bogdanov, Luca Trevisan

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, prompted by way of targeted purposes: the advancements of the rules of cryptography and the hunt for ways to "cope" with the intractability of NP-hard difficulties. This survey appears to be like at either, and customarily 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 desktop technological know-how. The reader also will find a variety of effects, insights, and facts recommendations whose usefulness is going past the learn of average-case complexity.

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Randomized errorless search) We say A is a randomized errorless search algorithm for (L, D), where L ∈ NP, if there is a polynomial p such that (1) For every n, δ > 0, A runs in time p(n/δ) and outputs either a string w or the special symbol ⊥. (2) For every n, δ > 0 and x ∈ L, PrA [A(x; n, δ) outputs a witness for x or A(x; n, δ) = ⊥] > 1/2. (3) For every n and δ > 0, Prx∼Dn PrA [A(x; n, δ) = ⊥] > 1/4 ≤ δ. This definition is robust with respect to the choice of constants 21 and 14 ; it would remain equivalent if 21 and 14 were replaced by any two constants c and c , respectively, where 0 < c < c < 1.

5 A distribution for which worst case and average case are equivalent In this section we show that there exists a (possibly non-samplable) ensemble of distributions with respect to which worst-case and average-case tractability are equivalent notions. Thus, the study of average-case complexity with respect to all ensembles reduces to the study of worst-case complexity, and in this sense it is natural to consider restricted classes such as computable and samplable ensembles, as we do in the remainder of this study.

Fix an x ∈ Supp Dn . If Dn (x) ≤ 2−|x| , then simply let C(x) = 0x, that is, 0 concatenated with x. 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 .

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