Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

Abstract

AbstractIn a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include: Certifying that a list of n integers has no 3-SUM solution can be done in Merlin–Arthur time $$\tilde{O}(n)$$ O ~ ( n ) . Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in $$\tilde{O}(n^{1.5})$$ O ~ ( n 1.5 ) time (that is, there is a proof system with proofs of length $$\tilde{O}(n^{1.5})$$ O ~ ( n 1.5 ) and a deterministic verifier running in $$\tilde{O}(n^{1.5})$$ O ~ ( n 1.5 ) time). Counting the number of k-cliques with total edge weight equal to zero in an n-node graph can be done in Merlin–Arthur time $${\tilde{O}}(n^{\lceil k/2\rceil })$$ O ~ ( n ⌈ k / 2 ⌉ ) (where $$k\ge 3$$ k ≥ 3 ). For odd k, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in an m-edge graph can be done in Merlin–Arthur time $${\tilde{O}}(m)$$ O ~ ( m ) . Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only count k-cliques in unweighted graphs, and had worse running times for small k. Computing the All-Pairs Shortest Distances matrix for an n-node graph can be done in Merlin–Arthur time $$\tilde{O}(n^2)$$ O ~ ( n 2 ) . Note this is optimal, as the matrix can have $$\Omega (n^2)$$ Ω ( n 2 ) nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an $$\tilde{O}(n^{2.94})$$ O ~ ( n 2.94 ) nondeterministic time algorithm. Certifying that an n-variable k-CNF is unsatisfiable can be done in Merlin–Arthur time $$2^{n/2 - n/O(k)}$$ 2 n / 2 - n / O ( k ) . We also observe an algebrization barrier for the previous $$2^{n/2}\cdot \textrm{poly}(n)$$ 2 n / 2 · poly ( n ) -time Merlin–Arthur protocol of R. Williams [CCC’16] for $$\#$$ # SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol for k-UNSAT running in $$2^{n/2}/n^{\omega (1)}$$ 2 n / 2 / n ω ( 1 ) time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol. Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time $$2^{4n/5}\cdot \textrm{poly}(n)$$ 2 4 n / 5 · poly ( n ) . Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in $$2^{2n/3}\cdot \textrm{poly}(n)$$ 2 2 n / 3 · poly ( n ) time. Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution to n integers can be done in Merlin–Arthur time $$2^{n/3}\cdot \textrm{poly}(n)$$ 2 n / 3 · poly ( n ) , improving on the previous best protocol by Nederlof [IPL 2017] which took $$2^{0.49991n}\cdot \textrm{poly}(n)$$ 2 0.49991 n · poly ( n ) time.