Distributed PCP Theorems for Hardness of Approximation in P
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)(2017)
摘要
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 x_1, …, x_n/2, Bob knows x_n/2+1,…,x_n, 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^(log n)^1-o(1); only (1+o(1))-factor lower bounds (under SETH) were known before.
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关键词
fine-grained complexity,similarity search,strong exponential-time hypothesis,closest pair,longest common subsequence,inapproximability
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