Distributed PCP Theorems for Hardness of Approximation in P
Amir Abboud, Aviad Rubinstein, Ryan Williams · 2017
We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment x ∈ {0, 1}nto 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 trulysubquadratic approximation algorithms for Maximum Inner Product over {0, 1}-vectors, LCS Closest Pair over permutations, Approximate Partial Match, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first three problems we obtain nearly-polynomial factors of 2(log n)1-o(1); only (1+o(1))-factor lower bounds (under SETH) were known before. As an additional feature of our reduction, we obtain new SETH lower bounds for the exact “monochromatic” Closest Pair problem in the Euclidean, Manhattan, and Hamming metrics.