On the Cryptographic Hardness of Local Search

We show new hardness results for the class of Polynomial Local Search problems (PLS): - Hardness of PLS based on a falsifiable assumption on bilinear groups introduced by Kalai, Paneth, and Yang (STOC 2019), and the Exponential Time Hypothesis for randomized algorithms. Previous standard model constructions relied on non-falsifiable and non-standard assumptions. - Hardness of PLS relative to random oracles. The construction is essentially different than previous constructions, and in particular is unconditionally secure. The construction also demonstrates the hardness of parallelizing local search. The core observation behind the results is that the unique proofs property of incrementally-verifiable computations previously used to demonstrate hardness in PLS can be traded with a simple incremental completeness property

New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method

In this paper, we obtain several new results on lower bounds and derandomization for ACC? circuits (constant-depth circuits consisting of AND/OR/MOD_m gates for a fixed constant m, a frontier class in circuit complexity): 1) We prove that any polynomial-time Merlin-Arthur proof system with an ACC? verifier (denoted by MA_{ACC?}) can be simulated by a nondeterministic proof system with quasi-polynomial running time and polynomial proof length, on infinitely many input lengths. This improves the previous simulation by [Chen, Lyu, and Williams, FOCS 2020], which requires both quasi-polynomial running time and proof length. 2) We show that MA_{ACC?} cannot be computed by fixed-polynomial-size ACC? circuits, and our hard languages are hard on a sufficiently dense set of input lengths. 3) We show that NEXP (nondeterministic exponential-time) does not have ACC? circuits of sub-half-exponential size, improving the previous sub-third-exponential size lower bound for NEXP against ACC? by [Williams, J. ACM 2014]. Combining our first and second results gives a conceptually simpler and derandomization-centric proof of the recent breakthrough result NQP := NTIME[2^polylog(n)] ? ? ACC? by [Murray and Williams, SICOMP 2020]: Instead of going through an easy witness lemma as they did, we first prove an ACC? lower bound for a subclass of MA, and then derandomize that subclass into NQP, while retaining its hardness against ACC?. Moreover, since our derandomization of MA_{ACC?} achieves a polynomial proof length, we indeed prove that nondeterministic quasi-polynomial-time with n^?(1) nondeterminism bits (denoted as NTIMEGUESS[2^polylog(n), n^?(1)]) has no poly(n)-size ACC? circuits, giving a new proof of a result by Vyas. Combining with a win-win argument based on randomized encodings from [Chen and Ren, STOC 2020], we also prove that NTIMEGUESS[2^polylog(n), n^?(1)] cannot be 1/2+1/poly(n)-approximated by poly(n)-size ACC? circuits, improving the recent strongly average-case lower bounds for NQP against ACC? by [Chen and Ren, STOC 2020]. One interesting technical ingredient behind our second result is the construction of a PSPACE-complete language that is paddable, downward self-reducible, same-length checkable, and weakly error correctable. Moreover, all its reducibility properties have corresponding AC?[2] non-adaptive oracle circuits. Our construction builds and improves upon similar constructions from [Trevisan and Vadhan, Complexity 2007] and [Chen, FOCS 2019], which all require at least TC? oracle circuits for implementing these properties

Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones

We show that a wide variety of non-linear cellular automata (CAs) can be decomposed into a quasidirect product of linear ones. These CAs can be predicted by parallel circuits of depth O(log^2 t) using gates with binary inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of inputs are allowed. Thus these CAs can be predicted by (idealized) parallel computers much faster than by explicit simulation, even though they are non-linear. This class includes any CA whose rule, when written as an algebra, is a solvable group. We also show that CAs based on nilpotent groups can be predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p'' gates respectively. We use these techniques to give an efficient algorithm for a CA rule which, like elementary CA rule 18, has diffusing defects that annihilate in pairs. This can be used to predict the motion of defects in rule 18 in O(log^2 t) parallel time

Satisfiability and Derandomization for Small Polynomial Threshold Circuits

A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth. - Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only. - Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1/c^d} wires such that C has at most 2^{n^{1-1/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity. - Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1/epsilon)

Quantified Derandomization of Linear Threshold Circuits

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC^0$, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for $TC^0$. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of $TC^0$ circuits of depth $d>2$. Our first main result is a quantified derandomization algorithm for $TC^0$ circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a $TC^0$ circuit $C$ over $n$ input bits with depth $d$ and $n^{1+\exp(-d)}$ wires, runs in almost-polynomial-time, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs. In fact, our algorithm works even when the circuit $C$ is a linear threshold circuit, rather than just a $TC^0$ circuit (i.e., $C$ is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of $TC^0$, and would consequently imply that $NEXP\not\subseteq TC^0$. Specifically, if there exists a quantified derandomization algorithm that gets as input a $TC^0$ circuit with depth $d$ and $n^{1+O(1/d)}$ wires (rather than $n^{1+\exp(-d)}$ wires), runs in time at most $2^{n^{\exp(-d)}}$, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs, then there exists an algorithm with running time $2^{n^{1-\Omega(1)}}$ for standard derandomization of $TC^0$.Comment: Changes in this revision: An additional result (a PRG for quantified derandomization of depth-2 LTF circuits); rewrite of some of the exposition; minor correction

Faster all-pairs shortest paths via circuit complexity

We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two $n \times n$ matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense $n$-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time $$\frac{n^3}{2^{\Omega(\log n)^{1/2}}}$$ and is correct with high probability. On the word RAM, the algorithm runs in $n^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log M$ time for edge weights in $([0,M] \cap {\mathbb Z})\cup\{\infty\}$. Prior algorithms used either $n^3/(\log^c n)$ time for various $c \leq 2$, or $O(M^{\alpha}n^{\beta})$ time for various $\alpha > 0$ and $\beta > 2$. The new algorithm applies a tool from circuit complexity, namely the Razborov-Smolensky polynomials for approximately representing ${\sf AC}^0[p]$ circuits, to efficiently reduce a matrix product over the $(\min,+)$ algebra to a relatively small number of rectangular matrix products over ${\mathbb F}_2$, each of which are computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in $n^3/2^{\log^{\delta} n}$ time for some $\delta > 0$, which utilizes the Yao-Beigel-Tarui translation of ${\sf AC}^0[m]$ circuits into "nice" depth-two circuits.Comment: 24 pages. Updated version now has slightly faster running time. To appear in ACM Symposium on Theory of Computing (STOC), 201

An average-case lower bound against ACC0

In a seminal work, Williams [22] showed that NEXP (nondeterministic exponential time) does not have polynomial-size ACC0 circuits. Williams’ technique inherently gives a worst-case lower bound, and until now, no average-case version of his result was known. We show that there is a language L in NEXP and a function ε(n)=1/ log(n) ω(1) such that no sequence of polynomial size ACC0 circuits solves L on more than a 1/2+ε(n) fraction of inputs of length n for all large enough n. Complementing this result, we give a nontrivial pseudo-random generator against polynomial-size AC0[6] circuits. We also show that learning algorithms for quasi-polynomial size ACC0 circuits running in time 2n/nω(1) imply lower bounds for the randomised exponential time classes RE (randomized time 2O(n) with one-sided error) and ZPE/1 (zero-error randomized time 2O(n) with 1 bit of advice) against polynomial size ACC0 circuits. This strengthens results of Oliveira and Santhanam [15]