Infeasibility of instance compression and succinct PCPs for NP

Abstract

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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