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

Abstract

In 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). Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in tilde{O}(n^{1.5}) time (that is, there is a proof system with proofs of length tilde{O}(n^{1.5}) and a deterministic verifier running in tilde{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/2rceil }) (where kge 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). 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). Note this is optimal, as the matrix can have Omega (n^2) nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an tilde{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)}. We also observe an algebrization barrier for the previous 2^{n/2}cdot textrm{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)} 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). 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) 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), improving on the previous best protocol by Nederlof [IPL 2017] which took 2^{0.49991n}cdot textrm{poly}(n) time.