Infeasibility of Instance Compression and Succinct PCPs for NP

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. [BDFH08] and Harnik and Naor [HN06b] 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 collision-resistant hash functions from one-way functions is unlikely to be viable in its present form. • (Buhrman-Hitchcock) There are no subexponential-size hard...

• Type: Journal
• Year: 2007
• Where: ECCC
• Authors: Lance Fortnow, Rahul Santhanam
• Keywords: Derandomization Hypothesis, ECCC 2007, OR-SAT Problem, Parametric Np Problems