Average-Case Complexity (Foundations and Trends(R) in 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 examine of the average-case complexity of intractable difficulties begun within the Seventies, influenced by way of detailed purposes: the advancements of the rules of cryptography and the quest for ways to "cope" with the intractability of NP-hard difficulties. This survey appears to be like 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 desktop technology. The reader also will find a variety of effects, insights, and facts strategies whose usefulness is going past the learn of average-case complexity.

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Abusing terminology, we sometimes call A a search algorithm for the NP-language LV consisting of all x for which such a witness w exists. Thus, when we say “a search algorithm for L” we mean an algorithm that on input x ∈ L outputs an NP-witness w that x is a member of L, with respect to an implicit NP-relation V such that L = LV . Designing search algorithms for languages in NP appears to be in general a harder task than designing decision algorithms. An efficient search algorithm for a language in NP immediately yields an efficient decision algorithm for the same language.

This establishes the claim and proves that (L, D) ∈ AvgP. 2. 2 37 The completeness result In this section we prove the existence of a complete problem for (NP, PComp), the class of all distributional problems (L, D) such that L is in NP and D is polynomial-time computable. Our problem is the following “bounded halting” problem for non-deterministic Turing machines: BH = {(M, x, 1t ) : M is a non-deterministic Turing machine that accepts x in ≤ t steps}. 1) Note that BH is NP-complete: Let L be a language in NP and M be a non-deterministic Turing machine that decides L in time at most p(n) on inputs of length n.

5 can be replaced with 1/nε for any fixed ε > 0. 4 Decision Versus Search and One-Way Functions In worst-case complexity, a search algorithm A for an NP-relation V is required to produce, on input x, a witness w of length poly(|x|) such that V accepts (x; w), whenever such a w exists. Abusing terminology, we sometimes call A a search algorithm for the NP-language LV consisting of all x for which such a witness w exists. Thus, when we say “a search algorithm for L” we mean an algorithm that on input x ∈ L outputs an NP-witness w that x is a member of L, with respect to an implicit NP-relation V such that L = LV .

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