Infeasibility of instance compression and succinct PCPs for NP

The OR-SAT problem asks, given Boolean formulae phi(1), ..., phi(m) each of size at most n, whether at least one of the phi(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 subset of 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 subset of coNP/poly. (ii) Satisfiability does not have PCPs of size polynomial in the number of variables unless NP subset of coNP/poly. (iii) An approach of Harnik 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. (C) 2010 Elsevier Inc. All rights reserved.