Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

We present an efficient proof system for MULTIPOINT ARITHMETIC CIRCUIT EVALUATION: for any arithmetic circuit C(x1, . . . ,xn) of size s and degree d over a field F, and any inputs a1, . . . ,aK ∈ Fn, • the Prover sends the Verifier the values C(a1), . . . ,C(aK) ∈ F and a proof of Õ(K ·d) length, and • the Verifier tosses poly(log(dK|F|/ε)) coins and can check the proof in about Õ(K · (n+d)+ s) time, with probability of error less than ε . For small degree d, this “Merlin-Arthur” proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in cn time (for various c < 2) for the Permanent, #Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of 0-1 linear programs. In general, the value of any polynomial in Valiant’s class VP can be certified faster than “exhaustive summation” over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in 22n/3+o(n) time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in nk/2+O(1)-time for counting k-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.