We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment x∈{0,1}n to a CNF formula φ is
shared between two parties, where Alice knows x1,…,xn/2, Bob knows
xn/2+1,…,xn, and both parties know φ. The goal is to have
Alice and Bob jointly write a PCP that x satisfies φ, while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
x.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of 2(logn)1−o(1); only
(1+o(1))-factor lower bounds (under SETH) were known before