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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 ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$ $\tilde{O}\left(n\right)$. Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$time).Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$ $\tilde{O}\left(n^{\lceil k/2\rceil }\right)$(where$$k\ge 3$$ $k\ge 3$). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$ $\tilde{O}\left(m\right)$. Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$ $\tilde{O}\left(n^2\right)$. Note this is optimal, as the matrix can have$$\Omega (n^2)$$ $\Omega \left(n^2\right)$nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$ $\tilde{O}\left(n^{2.94}\right)$nondeterministic time algorithm.Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$ $2^{n/2-n/O\left(k\right)}$. We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$ $2^{n/2}·\text{poly}\left(n\right)$-time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$ $\#$SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$ $2^{n/2}/n^{\omega \left(1\right)}$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$$2^{4n/5}\cdot \textrm{poly}(n)$$ $2^{4n/5}·\text{poly}\left(n\right)$. 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$$2^{2n/3}\cdot \textrm{poly}(n)$$ $2^{2n/3}·\text{poly}\left(n\right)$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 tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$ $2^{n/3}·\text{poly}\left(n\right)$, improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$ $2^{0.49991n}·\text{poly}\left(n\right)$time. 

For a complexity class $$C$$ and language $$L$$, a constructive separation of $$L\notin C$$ gives an efficient algorithm (also called a refuter) to findcounterexamples (bad inputs) for every $$C$$-algorithm attempting to decide $$L$$.We study the questions: Which lower bounds can be made constructive? What arethe consequences of constructive separations? We build a case thatconstructiveness serves as a dividing line between many weak lower bounds weknow how to prove, and strong lower bounds against $$P$$, $ZPP$, and $BPP$. Putanother way, constructiveness is the opposite of a complexity barrier: it is aproperty we want lower bounds to have. Our results fall into three broadcategories. 1. Our first set of results shows that, for many well-known lower boundsagainst streaming algorithms, one-tape Turing machines, and query complexity,as well as lower bounds for the Minimum Circuit Size Problem, making theselower bounds constructive would imply breakthrough separations ranging from$$EXP \neq BPP$$ to even $$P \neq NP$$. 2. Our second set of results shows that for most major open problems in lowerbounds against $$P$$, $ZPP$, and $BPP$, including $$P \neq NP$$, $$P \neq PSPACE$$,$$P \neq PP$$, $$ZPP \neq EXP$$, and $$BPP \neq NEXP$$, any proof of the separationwould further imply a constructive separation. Our results generalize earlierresults for $$P \neq NP$$ [Gutfreund, Shaltiel, and Ta-Shma, CCC 2005] and $$BPP\neq NEXP$$ [Dolev, Fandina and Gutfreund, CIAC 2013]. 3. Our third set of results shows that certain complexity separations cannotbe made constructive. We observe that for all super-polynomially growingfunctions $$t$$, there are no constructive separations for detecting high$$t$$-time Kolmogorov complexity (a task which is known to be not in $$P$$) fromany complexity class, unconditionally. 

A fundamental problem in circuit complexity is to find explicit functions that require large depth to compute. When considering the natural DeMorgan basis of {OR,AND}, where negations incur no cost, the best known depth lower bounds for an explicit function in NP have the form (3-o(1))log₂ n, established by Håstad (building on others) in the early 1990s. We make progress on the problem of improving this factor of 3, in two different ways: - We consider an "algorithmic method" approach to proving stronger depth lower bounds for non-uniform circuits in the DeMorgan basis. We show that slightly faster algorithms (than what is known) for counting the number of satisfying assignments on subcubic-size DeMorgan formulas would imply supercubic-size DeMorgan formula lower bounds, implying that the depth must be at least (3+ε)log₂ n for some ε > 0. For example, if #SAT on formulas of size n^{2+2ε} can be solved in 2^{n - n^{1-ε}log^k n} time for some ε > 0 and a sufficiently large constant k, then there is a function computable in 2^{O(n)} time with a SAT oracle which does not have n^{3+ε}-size formulas. In fact, the #SAT algorithm only has to work on formulas that are a conjunction of n^{1-ε} subformulas, each of which is n^{1+3ε} size, in order to obtain the supercubic lower bound. As a proof of concept, we show that our new algorithms-to-lower-bounds connection can be applied to prove new lower bounds for "hybrid" DeMorgan formula models which compute interesting functions at their leaves. - Turning to the {NAND} basis, we establish a greater-than-(3 log₂ n) depth lower bound against uniform circuits solving the SAT problem, using an extension of the "indirect diagonalization" method for NAND formulas. Note that circuits over the NAND basis are a special case of circuits over the DeMorgan basis; however, hard functions such as Andreev’s function (known to require depth (3-o(1))log₂ n in the DeMorgan basis) can still be computed with NAND circuits of depth (3+o(1))log₂ n. Our results imply that SAT requires polylogtime-uniform NAND circuits of depth at least 3.603 log₂ n. 

Given the need for ever higher performance, and the failure of CPUs to keep providing single-threaded performance gains, engineers are increasingly turning to highly-parallel custom VLSI chips to implement expensive computations. In VLSI design, the gates and wires of a logical circuit are placed on a 2-dimensional chip with a small number of layers. Traditional VLSI models use gate delay to measure the time complexity of the chip, ignoring the lengths of wires. However, as technology has advanced, wire delay is no longer negligible; it has become an important measure in the design of VLSI chips [Markov, Nature (2014)]. Motivated by this situation, we define and study a model for VLSI chips, called wire-delay VLSI, which takes wire delay into account, going beyond an earlier model of Chazelle and Monier [JACM 1985]. - We prove nearly tight upper bounds and lower bounds (up to logarithmic factors) on the time delay of this chip model for several basic problems. For example, And, Or and Parity require Θ(n^{1/3}) delay, while Addition and Multiplication require ̃ Θ(n^{1/2}) delay, and Triangle Detection on (dense) n-node graphs requires ̃ Θ(n) delay. Interestingly, when we allow input bits to be read twice, the delay for Addition can be improved to Θ(n^{1/3}). - We also show that proving significantly higher lower bounds in our wire-delay VLSI model would imply breakthrough results in circuit lower bounds. Motivated by this barrier, we also study conditional lower bounds on the delay of chips based on the Orthogonal Vectors Hypothesis from fine-grained complexity.