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Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases when the distance between the distributions is small (less than max(We also give an Ω(n 2/3 −2/3 ) lower bound. Our algorithm has applications to the problem of checking whether a given Markov process is rapidly mixing. We develop sublinear algorithms for this problem as well. * A preliminary version of this

A new algebraic technique for the construction of interactive proof systems is presented. Our technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP = PSPACE [28] and that MIP = NEXP [4].

The OR-SAT problem asks, given Boolean formulae φ 1 , . . . , φ m each of size at most n, whether at least one of the φ i 's is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the Polynomial-Time Hierarchy collapses. This result settles an open problem proposed by Bodlaender et al. (2008) [6] and Harnik and Naor (2006) [20] and has a number of implications. (i) A number of parametric NP problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP/poly. (ii) Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly. (iii) An approach ofHarnik and Naor to constructing collision-resistant hash functions from one-way functions is unlikely to be viable in its present form. (iv) (Buhrman-Hitchcock) There are no subexponential-size hard sets for NP unless NP is in co-NP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions.

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It's one of the fundamental mathematical problems of our time, and its importance grows with the rise of powerful computers.