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.

Abstract-We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search.Our first main result is a simple deterministic algorithm running in time 2 n−Ω(n) for satisfiability of formulae of linear size in n, where n is the number of variables in the formula. This algorithm extends to exactly counting the number of satisfying assignments, within the same time bound.Our second main result is a deterministic algorithm running in time 2 n−Ω(n/ log(n)) for solving QBFs in which the number of occurrences of any variable is bounded by a constant. For instances which are "structured", in a certain precise sense, the algorithm can be modified to run in time 2 n−Ω(n) . To the best of our knowledge, no non-trivial algorithms were known for these problems before.As a byproduct of the technique used to establish our first main result, we show that every function computable by linear-size formulae can be represented by decision trees of size 2 n−Ω(n) . As a consequence, we get strong superlinear average-case formula size lower bounds for the Parity function.

The OR-SAT problem asks, given Boolean formulae φ1, . . . , φm each of size at most n, whether at least one of the φi'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. [4] and Harnik and Naor [15] and has a number of implications.• 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.• Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly.• An approach of Harnik and Naor to constructing collisionresistant hash functions from one-way functions is unlikely to be viable in its present form.• (Buhrman-Hitchcock) There are no subexponentialsize 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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