Abstract
We provide a complete picture of the extent to which amplification of success probability is possible for randomized algorithms having access to one NP oracle query, in the settings of two-sided, onesided, and zero-sided error. We generalize this picture to amplifying one-query algorithms with q-query algorithms, and we show our inclusions are tight for relativizing techniques.
Highlights
Amplification of the success probability of randomized algorithms is a ubiquitous tool in complexity theory
We investigate amplification for randomized reductions to NP-complete problems, which can be modeled as randomized algorithms with the ability to make queries to an NP oracle
The usual amplification strategy involves running multiple independent trials, which would increase the number of NP oracle queries, so this does not generally work if we restrict the number of queries
Summary
Amplification of the success probability of randomized algorithms is a ubiquitous tool in complexity theory. The best strategy for amplifying plain randomized algorithms is to take the majority vote of q independent trials, which in our setting would naively involve q NP oracle queries. For ∈ (0, 1] (the advantage), BPPNP[1] is the set of all languages solvable by a randomized polynomial-time algorithm that may make one query to an NP oracle and produces the correct output with probability. 1 k+1 where an integer, Theorem tells us the best advantage achievable with q nonadaptive NP queries using relativizing techniques: if k is even we can amplify to essentially q k. We point out that none of the inclusions in this paper can be strengthened to yield advantage exactly 1 via relativizing techniques, since BPP ⊆ ZPPN>P1[/12] relativizes [2] but BPP ⊆ PNP relative to an oracle [folklore]
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