Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > 2021:
All reports in year 2021:
TR21-138 | 23rd September 2021
Rahul Santhanam, Iddo Tzameret

#### Iterated Lower Bound Formulas: A Diagonalization-Based Approach to Proof Complexity

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally.

We give an explicit sequence of CNF formulas $\{\phi_n\}$ such that VNP$\neq$VP iff there are ... more >>>

TR21-137 | 14th September 2021
Xuangui Huang, Peter Ivanov, Emanuele Viola

#### Affine extractors and AC0-Parity

We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. ... more >>>

TR21-136 | 13th September 2021
Gil Cohen, Tal Yankovitz

#### LCC and LDC: Tailor-made distance amplification and a refined separation

The Alon-Edmonds-Luby distance amplification procedure (FOCS 1995) is an algorithm that transforms a code with vanishing distance to a code with constant distance. AEL was invoked by Kopparty, Meir, Ron-Zewi, and Saraf (J. ACM 2017) for obtaining their state-of-the-art LDC, LCC and LTC. Cohen and Yankovitz (CCC 2021) devised a ... more >>>

TR21-135 | 6th September 2021
Olaf Beyersdorff, Joshua Blinkhorn, Tomáš Peitl

#### Strong (D)QBF Dependency Schemes via Implication-free Resolution Paths

We suggest a general framework to study dependency schemes for dependency quantified Boolean formulas (DQBF). As our main contribution, we exhibit a new infinite collection of implication-free DQBF dependency schemes that generalise the reflexive resolution path dependency scheme. We establish soundness of all these schemes, implying that they can be ... more >>>

TR21-134 | 19th August 2021
Siddharth Bhaskar

#### A refinement of the Meyer-McCreight Union Theorem

For a function $t : 2^\star \to 1^\star$, let $C_t$ be the set of problems decidable on input $x$ in time at most $t(x)$ almost everywhere. The Union Theorem of Meyer and McCreight asserts that any union $\bigcup_{i < \omega} C_{t_i}$ for a uniformly recursive sequence of bounds $t_i$ is ... more >>>

TR21-133 | 12th September 2021
Oded Goldreich, Dana Ron

#### Testing Distributions of Huge Objects

Revisions: 1

We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings).
Accordingly, the ... more >>>

TR21-132 | 11th September 2021
Eric Binnendyk

#### Pseudo-random functions and uniform learnability

Boolean circuits are a model of computation. A class of Boolean circuits is called a polynomial class if the number of nodes is bounded by a polynomial function of the number of input variables. A class $C_n[s(n)]$ of Boolean functions is called learnable if there are algorithms that can approximate ... more >>>

TR21-131 | 10th September 2021
Olaf Beyersdorff, Benjamin Böhm

#### QCDCL with Cube Learning or Pure Literal Elimination - What is best?

Quantified conflict-driven clause learning (QCDCL) is one of the main approaches for solving quantified Boolean formulas (QBF). We formalise and investigate several versions of QCDCL that include cube learning and/or pure-literal elimination, and formally compare the resulting solving models via proof complexity techniques. Our results show that almost all of ... more >>>

TR21-130 | 7th September 2021
Srinivasan Arunachalam, João F. Doriguello

#### Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case

Hypercontractive inequalities for real-valued functions over the Boolean cube play an important role in theoretical computer science. In this work, we prove a hypercontractive inequality for matrix-valued functions defined over large alphabets, generalizing the result of Ben-Aroya, Regev, de Wolf (FOCS'08) for the Boolean alphabet. To obtain our result we ... more >>>

TR21-129 | 6th September 2021
Oded Goldreich, Dana Ron

#### A Lower Bound on the Complexity of Testing Grained Distributions

A distribution is called $m$-grained if each element appears with probability that is an integer multiple of $1/m$.
We prove that, for any constant $c<1$, testing whether a distribution over $[\Theta(m)]$ is $m$-grained requires $\Omega(m^c)$ samples.

more >>>

TR21-128 | 4th September 2021
Xiaotie Deng, Yuhao Li, David Mguni, Jun Wang, Yaodong Yang

#### On the Complexity of Computing Markov Perfect Equilibrium in General-Sum Stochastic Games

Similar to the role of Markov decision processes in reinforcement learning, Markov Games (also called Stochastic Games)lay down the foundation for the study of multi-agent reinforcement learning (MARL) and sequential agent interactions. In this paper, we introduce the solution concept, approximate Markov Perfect Equilibrium (MPE), to finite-state Stochastic Games repeated ... more >>>

TR21-127 | 30th August 2021
Ron D. Rothblum, Michael Ezra

#### Small Circuits Imply Efficient Arthur-Merlin Protocols

The inner product function $\langle x,y \rangle = \sum_i x_i y_i \bmod 2$ can be easily computed by a (linear-size) ${AC}^0(\oplus)$ circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on ... more >>>

TR21-126 | 25th August 2021
Yilei Chen, Qipeng Liu, Mark Zhandry

#### Quantum Algorithms for Variants of Average-Case Lattice Problems via Filtering

We show polynomial-time quantum algorithms for the following problems:
(*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a ... more >>>

TR21-125 | 23rd August 2021
Zhiyuan Fan, Jiatu Li, Tianqi Yang

#### The Exact Complexity of Pseudorandom Functions and Tight Barriers to Lower Bound Proofs

How much computational resource do we need for cryptography? This is an important question of both theoretical and practical interests. In this paper, we study the problem on pseudorandom functions (PRFs) in the context of circuit complexity. Perhaps surprisingly, we prove extremely tight upper and lower bounds in various circuit ... more >>>

TR21-124 | 17th August 2021
Iftach Haitner, Noam Mazor, Jad Silbak, Eliad Tsfadia

#### On the Complexity of Two-Party Differential Privacy

Revisions: 1

In distributed differential privacy, the parties perform analysis over their joint data while preserving the privacy for both datasets. Interestingly, for a few fundamental two-party functions such as inner product and Hamming distance, the accuracy of the distributed solution lags way behind what is achievable in the client-server setting. McGregor, ... more >>>

TR21-123 | 25th August 2021
Ben Davis, Hamed Hatami, William Pires, Ran Tao, Hamza Usmani

#### On public-coin zero-error randomized communication complexity

Revisions: 1

We prove that for every Boolean function, the public-coin zero-error randomized communication complexity and the deterministic communication complexity are polynomially equivalent.

more >>>

TR21-122 | 24th August 2021
Sabyasachi Basu, Akash Kumar, C. Seshadhri

#### The complexity of testing all properties of planar graphs, and the role of isomorphism

Consider property testing on bounded degree graphs and let $\varepsilon > 0$ denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on $\varepsilon$. Recent advances in testing minor-freeness have proven that ... more >>>

TR21-121 | 21st August 2021
Sumanta Ghosh, Rohit Gurjar

#### Matroid Intersection: A pseudo-deterministic parallel reduction from search to weighted-decision

We study the matroid intersection problem from the parallel complexity perspective. Given
two matroids over the same ground set, the problem asks to decide whether they have a common base and its search version asks to find a common base, if one exists. Another widely studied variant is the weighted ... more >>>

TR21-120 | 18th August 2021
Roei Tell

#### How to Find Water in the Ocean: A Survey on Quantified Derandomization

The focus of this survey is the question of *quantified derandomization*, which was introduced by Goldreich and Wigderson (2014): Does derandomization of probabilistic algorithms become easier if we only want to derandomize algorithms that err with extremely small probability? How small does this probability need to be in order for ... more >>>

TR21-119 | 13th August 2021
Omar Alrabiah, Venkatesan Guruswami

#### Visible Rank and Codes with Locality

Revisions: 1

We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call "visible rank." The locality constraints of a linear code are stipulated by a matrix $H$ of $\star$'s and $0$'s (which we ... more >>>

TR21-118 | 11th August 2021
Daniel Augot, Sarah Bordage, Jade Nardi

#### Efficient multivariate low-degree tests via interactive oracle proofs of proximity for polynomial codes

We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an
oracle function $f : L^m \rightarrow \mathbb F_q$, where $L\subset \mathbb F_q$, a verifier distinguishes whether $f$ is the evaluation of a ... more >>>

TR21-117 | 11th August 2021
Divesh Aggarwal, Bhavana Kanukurthi, SaiLakshmiBhavana Obbattu, Maciej Obremski, Sruthi Sekar

#### Simplicity Meets Near-Optimal Rate: Non-malleable Codes and Non-malleable Two-source Extractors via Rate Boosters

Revisions: 2

At ITCS 2010, Dziembowski, Pietrzak, and Wichs introduced Non-malleable Codes (NMCs). Non-malleability is one of the strongest and most challenging notions of security considered in cryptography and protects against tampering attacks. In the context of coding schemes, non-malleability requires that it be infeasible to tamper the codeword of a message ... more >>>

TR21-116 | 10th August 2021
Nai-Hui Chia, Chi-Ning Chou, Jiayu Zhang, Ruizhe Zhang

#### Quantum Meets the Minimum Circuit Size Problem

Revisions: 1

In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory---its hardness is mysterious, and a better understanding of its hardness can have ... more >>>

TR21-115 | 6th August 2021
Scott Aaronson, Andris Ambainis, Andrej Bogdanov, Krishnamoorthy Dinesh, Cheung Tsun Ming

#### On quantum versus classical query complexity

Revisions: 2

Aaronson and Ambainis (STOC 2015, SICOMP 2018) claimed that the acceptance probability of every quantum algorithm that makes $q$ queries to an $N$-bit string can be estimated to within $\epsilon$ by a randomized classical algorithm of query complexity $O_q((N/\epsilon^2)^{1-1/2q})$. We describe a flaw in their argument but prove that the ... more >>>

TR21-114 | 29th July 2021
Henning Fernau, Kshitij Gajjar

#### The Space Complexity of Sum Labelling

A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider ... more >>>

TR21-113 | 25th July 2021
Nikhil Mande, Ronald de Wolf

#### Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $\epsilon$, getting the optimal constant factors in the leading terms in a number of different models.

The following are our results in the randomized model:

1) We give a general technique to convert ... more >>>

TR21-112 | 30th July 2021
Vikraman Arvind, Venkatesan Guruswami

#### CNF Satisfiability in a Subspace and Related Problems

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of ... more >>>

TR21-111 | 19th July 2021
Aniruddha Biswas, Palash Sarkar

#### Influence of a Set of Variables on a Boolean Function

The influence of a set of variables on a Boolean function has three separate definitions in the literature, the first due to Ben-Or and Linial (1989), the second due to Fischer et al. (2002) and Blais (2009) and the third due to Tal (2017). The goal of the present work ... more >>>

TR21-110 | 22nd July 2021
Jaroslaw Blasiok, Peter Ivanov, Yaonan Jin, Chin Ho Lee, Rocco Servedio, Emanuele Viola

#### Fourier growth of structured $\mathbb{F}_2$-polynomials and applications

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of "structured" $\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at ... more >>>

TR21-109 | 20th July 2021
Sravanthi Chede, Anil Shukla

#### QRAT Polynomially Simulates Merge Resolution.

Merge Resolution (MRes [Beyersdorff et al. J. Autom. Reason.'2021] ) is a refutational proof system for quantified Boolean formulas (QBF). Each line of MRes consists of clauses with only existential literals, together with information of countermodels stored as merge maps. As a result, MRes has strategy extraction by design. The ... more >>>

TR21-108 | 22nd July 2021
Edward Pyne, Salil Vadhan

#### Limitations of the Impagliazzo--Nisan--Wigderson Pseudorandom Generator against Permutation Branching Programs

The classic Impagliazzo--Nisan--Wigderson (INW) psesudorandom generator (PRG) (STOC 94) for space-bounded computation uses a seed of length $O(\log n \cdot \log(nwd/\varepsilon))$ to fool ordered branching programs of length $n$, width $w$, and alphabet size $d$ to within error $\varepsilon$. A series of works have shown that the analysis of the ... more >>>

TR21-107 | 20th July 2021
igor razgon

#### Classification of OBDD size for monotone 2-CNFs

We introduce a new graph parameter called linear upper maximum induced
matching width \textsc{lu-mim width}, denoted for a graph $G$ by $lu(G)$.
We prove that the smallest size of the \textsc{obdd} for $\varphi$,
the monotone 2-\textsc{cnf} corresponding to $G$, is sandwiched
between $2^{lu(G)}$ and $n^{O(lu(G))}$.
The upper bound ... more >>>

TR21-106 | 22nd July 2021
Eshan Chattopadhyay, Jesse Goodman, David Zuckerman

#### The Space Complexity of Sampling

Recently, there has been exciting progress in understanding the complexity of distributions. Here, the goal is to quantify the resources required to generate (or sample) a distribution. Proving lower bounds in this new setting is more challenging than in the classical setting, and has yielded interesting new techniques and surprising ... more >>>

TR21-105 | 22nd July 2021
Suryajith Chillara

#### Functional lower bounds for restricted arithmetic circuits of depth four

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d} \in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then ... more >>>

TR21-104 | 26th June 2021
Sravanthi Chede, Anil Shukla

#### Does QRAT simulate IR-calc? QRAT simulation algorithm for $\forall$Exp+Res cannot be lifted to IR-calc

We show that the QRAT simulation algorithm of $\forall$Exp+Res from [B. Kiesl and M. Seidl, 2019] cannot be lifted to IR-calc.

more >>>

TR21-103 | 18th July 2021
Eli Ben-Sasson, Dan Carmon, Swastik Kopparty, David Levit

#### Elliptic Curve Fast Fourier Transform (ECFFT) Part I: Fast Polynomial Algorithms over all Finite Fields

Over finite fields $F_q$ containing a root of unity of smooth order $n$ (smoothness means $n$ is the product of small primes), the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division, interpolation and multi-point evaluation. These operations can ... more >>>

TR21-102 | 13th July 2021
Siddharth Iyer, Anup Rao, Victor Reis, Thomas Rothvoss, Amir Yehudayoff

Revisions: 1

We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\| \hat{f}_\ell \|_1$ is bounded by $d^\ell e^{{\ell+1 \choose 2}} ... more >>> TR21-101 | 13th July 2021 Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, Wei Zhan #### A Parallel Repetition Theorem for the GHZ Game: A Simpler Proof Revisions: 1 We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the$n$-fold GHZ game is at most$n^{-\Omega(1)}$. This was first established by Holmgren and ... more >>> TR21-100 | 11th July 2021 Christian Ikenmeyer, Balagopal Komarath, Nitin Saurabh #### Karchmer-Wigderson Games for Hazard-free Computation We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we ... more >>> TR21-099 | 4th July 2021 Louis Golowich #### Improved Product-Based High-Dimensional Expanders High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary combinatorial construction with near-optimal expansion is known. In this paper, we introduce an improved combinatorial high-dimensional expander construction, by modifying ... more >>> TR21-098 | 7th July 2021 Srikanth Srinivasan, S Venkitesh #### On the Probabilistic Degree of an$n$-variate Boolean Function Nisan and Szegedy (CC 1994) showed that any Boolean function$f:\{0,1\}^n\to\{0,1\}$that depends on all its input variables, when represented as a real-valued multivariate polynomial$P(x_1,\ldots,x_n)$, has degree at least$\log n - O(\log \log n)$. This was improved to a tight$(\log n - O(1))$bound by Chiarelli, Hatami ... more >>> TR21-097 | 7th July 2021 Jacobo Toran, Florian Wörz #### Number of Variables for Graph Identification and the Resolution of GI Formulas We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and positive depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution. Using this connection, we ... more >>> TR21-096 | 8th July 2021 Boaz Menuhin, Moni Naor #### Keep That Card in Mind: Card Guessing with Limited Memory A card guessing game is played between two players, Guesser and Dealer. At the beginning of the game, the Dealer holds a deck of$n$cards (labeled$1, ..., n$). For$n$turns, the Dealer draws a card from the deck, the Guesser guesses which card was drawn, and then ... more >>> TR21-095 | 8th July 2021 Marco Carmosino, Valentine Kabanets, Antonina Kolokolova, Igor Oliveira #### LEARN-Uniform Circuit Lower Bounds and Provability in Bounded Arithmetic We investigate randomized LEARN-uniformity, which captures the power of randomness and equivalence queries (EQ) in the construction of Boolean circuits for an explicit problem. This is an intermediate notion between P-uniformity and non-uniformity motivated by connections to learning, complexity, and logic. Building on a number of techniques, we establish the ... more >>> TR21-094 | 6th July 2021 Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas #### New Non-FPT Lower Bounds for Some Arithmetic Formulas An Algebraic Formula for a polynomial$P\in F[x_1,\ldots,x_N]$is an algebraic expression for$P(x_1,\ldots,x_N)$using variables, field constants, additions and multiplications. Such formulas capture an algebraic analog of the Boolean complexity class$\mathrm{NC}^1.$Proving lower bounds against this model is thus an important problem. It is known that, to prove ... more >>> TR21-093 | 1st July 2021 Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, Akshayaram Srinivasan #### Bounded Indistinguishability for Simple Sources A pair of sources$\mathbf{X},\mathbf{Y}$over$\{0,1\}^n$are$k$-indistinguishable if their projections to any$k$coordinates are identically distributed. Can some$\mathit{AC^0}$function distinguish between two such sources when$k$is big, say$k=n^{0.1}$? Braverman's theorem (Commun. ACM 2011) implies a negative answer when$\mathbf{X}$is uniform, whereas Bogdanov et ... more >>> TR21-092 | 28th June 2021 Yanyi Liu, Rafael Pass #### A Note on One-way Functions and Sparse Languages We show equivalence between the existence of one-way functions and the existence of a \emph{sparse} language that is hard-on-average w.r.t. some efficiently samplable high-entropy'' distribution. In more detail, the following are equivalent: - The existentence of a$S(\cdot)$-sparse language$L$that is hard-on-average with respect to some samplable ... more >>> TR21-091 | 29th June 2021 Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, Amnon Ta-Shma #### Expander Random Walks: The General Case and Limitations Cohen, Peri and Ta-Shma (STOC'21) considered the following question: Assume the vertices of an expander graph are labelled by$\pm 1$. What "test" functions$f : \{\pm 1\}^t \to \{\pm1 \}$can or cannot distinguish$t$independent samples from those obtained by a random walk? [CPTS'21] considered only balanced labelling, ... more >>> TR21-090 | 14th June 2021 Divesh Aggarwal, Eldon Chung, Maciej Obremski, Joao Ribeiro #### On Secret Sharing, Randomness, and Random-less Reductions for Secret Sharing Secret-sharing is one of the most basic and oldest primitives in cryptography, introduced by Shamir and Blakely in the 70s. It allows to strike a meaningful balance between availability and confidentiality of secret information. It has a host of applications most notably in threshold cryptography and multi-party computation. All known ... more >>> TR21-089 | 25th June 2021 Hanlin Ren, Rahul Santhanam #### A Relativization Perspective on Meta-Complexity Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where: * MCSP can be solved in deterministic polynomial time, but ... more >>> TR21-088 | 23rd June 2021 Oded Goldreich #### Open Problems in Property Testing of Graphs Revisions: 1 We briefly discuss a few open problems in the study of various models of testing graph properties, focusing on the query complexity of the various tasks. In the dense graph model, we discuss several open problems, including: * Determining the complexity of testing triangle-freeness. * Characterizing the class of properties ... more >>> TR21-087 | 22nd June 2021 Uma Girish, Ran Raz #### Eliminating Intermediate Measurements using Pseudorandom Generators Revisions: 1 We show that quantum algorithms of time T and space$S \ge \log T$with intermediate measurements can be simulated by quantum algorithms of time$T\cdot \mathrm{poly}(S)$and space$O(S\cdot \log T)$without intermediate measurements. The best simulations prior to this work required either$\Omega(T)$space (by the deferred measurement ... more >>> TR21-086 | 22nd June 2021 Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Ameya Velingker, Santhoshini Velusamy #### Linear Space Streaming Lower Bounds for Approximating CSPs We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on$n$variables taking values in$\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of$q$requires$\Omega(n)$space even on instances with$O(n)$constraints. We also identify ... more >>> TR21-085 | 21st June 2021 Ilya Volkovich #### The Final Nail in the Coffin of Statistically-Secure Obfuscator. We present an elementary, self-contained proof of the result of Goldwasser and Rothblum [GR07] that the existence of a (perfect) statistically secure obfuscator implies a collapse of the polynomial hierarchy. In fact, we show that an existence of a weaker object implies a somewhat stronger statement. In addition, we extend ... more >>> TR21-084 | 21st June 2021 Liron Bronfman, Ron Rothblum #### PCPs and Instance Compression from a Cryptographic Lens Modern cryptography fundamentally relies on the assumption that the adversary trying to break the scheme is computationally bounded. This assumption lets us construct cryptographic protocols and primitives that are known to be impossible otherwise. In this work we explore the effect of bounding the adversary's power in other information theoretic ... more >>> TR21-083 | 21st June 2021 Mark Braverman, Sumegha Garg, Or Zamir #### Tight Space Complexity of the Coin Problem In the coin problem we are asked to distinguish, with probability at least$2/3$, between$ni.i.d.$coins which are heads with probability$\frac{1}{2}+\beta$from ones which are heads with probability$\frac{1}{2}-\beta$. We are interested in the space complexity of the coin problem, corresponding to the width of a read-once ... more >>> TR21-082 | 16th June 2021 Rahul Ilango, Hanlin Ren, Rahul Santhanam #### Hardness on any Samplable Distribution Suffices: New Characterizations of One-Way Functions by Meta-Complexity We show that one-way functions exist if and only if there is some samplable distribution D such that it is hard to approximate the Kolmogorov complexity of a string sampled from D. Thus we characterize the existence of one-way functions by the average-case hardness of a natural \emph{uncomputable} problem on ... more >>> TR21-081 | 14th June 2021 Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas #### Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits Revisions: 1 An Algebraic Circuit for a polynomial$P\in F[x_1,\ldots,x_N]$is a computational model for constructing the polynomial$P$using only additions and multiplications. It is a \emph{syntactic} model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to ... more >>> TR21-080 | 10th June 2021 Lijie Chen, Roei Tell #### Hardness vs Randomness, Revised: Uniform, Non-Black-Box, and Instance-Wise We propose a new approach to the hardness-to-randomness framework and to the promise-BPP=promise-P conjecture. Classical results rely on non-uniform hardness assumptions to construct derandomization algorithms that work in the worst-case, or rely on uniform hardness assumptions to construct derandomization algorithms that work only in the average-case. In both types of ... more >>> TR21-079 | 9th June 2021 Venkatesan Guruswami, Xiaoyu He, Ray Li #### The zero-rate threshold for adversarial bit-deletions is less than 1/2 We prove that there exists an absolute constant$\delta>0$such any binary code$C\subset\{0,1\}^N$tolerating$(1/2-\delta)N$adversarial deletions must satisfy$|C|\le 2^{\poly\log N}$and thus have rate asymptotically approaching$0$. This is the first constant fraction improvement over the trivial bound that codes tolerating$N/2$adversarial deletions must have rate ... more >>> TR21-078 | 8th June 2021 Rahul Jain, Srijita Kundu #### A direct product theorem for quantum communication complexity with applications to device-independent QKD We give a direct product theorem for the entanglement-assisted interactive quantum communication complexity of an$l$-player predicate$V$. In particular we show that for a distribution$p$that is product across the input sets of the$l$players, the success probability of any entanglement-assisted quantum communication protocol for computing$n$... more >>> TR21-077 | 6th June 2021 Shir Peleg, Amir Shpilka, Ben Lee Volk #### Lower Bounds on Stabilizer Rank The stabilizer rank of a quantum state$\psi$is the minimal$r$such that$\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$for$c_j \in \mathbb{C}$and stabilizer states$\varphi_j$. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the ... more >>> TR21-076 | 4th June 2021 Dmitry Sokolov #### Pseudorandom Generators, Resolution and Heavy Width Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson \cite{ABRW04} we call a pseudorandom generator$\mathrm{PRG}\colon \{0, 1\}^n \to \{0, 1\}^m$hard for for a propositional proof system$\mathrm{P}$if$\mathrm{P}$cannot efficiently prove the (properly encoded) statement$b \notin \mathrm{Im}(\mathrm{PRG})$for any string$b \in \{0, 1\}^m$. In \cite{ABRW04} authors ... more >>> TR21-075 | 4th June 2021 Eshan Chattopadhyay, Jesse Goodman, Jyun-Jie Liao #### Affine Extractors for Almost Logarithmic Entropy We give an explicit construction of an affine extractor (over$\mathbb{F}_2$) that works for affine sources on$n$bits with min-entropy$k \ge~ \log n \cdot (\log \log n)^{1 + o(1)}$. This improves prior work of Li (FOCS'16) that requires min-entropy at least$\mathrm{poly}(\log n)$. Our construction is ... more >>> TR21-074 | 3rd June 2021 Theodoros Papamakarios, Alexander Razborov #### Space characterizations of complexity measures and size-space trade-offs in propositional proof systems We identify two new big clusters of proof complexity measures equivalent up to polynomial and$\log n$factors. The first cluster contains, among others, the logarithm of tree-like resolution size, regularized (that is, multiplied by the logarithm of proof length) clause and monomial space, and clause space, both ordinary and ... more >>> TR21-073 | 3rd June 2021 Emanuele Viola #### Lower bounds for samplers and data structures via the cell-probe separator Suppose that a distribution$S$can be approximately sampled by an efficient cell-probe algorithm. It is shown to be possible to restrict the input to the algorithm so that its output distribution is still not too far from$S$, and at the same time many output coordinates are almost pairwise ... more >>> TR21-072 | 23rd May 2021 Pranjal Dutta, Gorav Jindal, Anurag Pandey, Amit Sinhababu #### Arithmetic Circuit Complexity of Division and Truncation Given polynomials$f,g,h\,\in \mathbb{F}[x_1,\ldots,x_n]$such that$f=g/h$, where both$g$and$h$are computable by arithmetic circuits of size$s$, we show that$f$can be computed by a circuit of size$\poly(s,\deg(h))$. This solves a special case of division elimination for high-degree circuits (Kaltofen'87 \& WACT'16). The result ... more >>> TR21-071 | 16th May 2021 Samuel Epstein #### On the Algorithmic Content of Quantum Measurements We show that given a quantum measurement, for an overwhelming majority of pure states, no meaningful information is produced. This is independent of the number of outcomes of the quantum measurement. Due to conservation inequalities, such random noise cannot be processed into coherent data. more >>> TR21-070 | 13th May 2021 Shuo Pang #### SOS lower bound for exact planted clique We prove a SOS degree lower bound for the planted clique problem on Erd{\"o}s-R\'enyi random graphs$G(n,1/2)$. The bound we get is degree$d=\Omega(\epsilon^2\log n/\log\log n)$for clique size$\omega=n^{1/2-\epsilon}$, which is almost tight. This improves the result of \cite{barak2019nearly} on the soft'' version of the problem, where the family ... more >>> TR21-069 | 12th May 2021 Dominik Scheder #### PPSZ is better than you think PPSZ, for long time the fastest known algorithm for k-SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to 0 or 1, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being ... more >>> TR21-068 | 8th May 2021 Marcel Dall'Agnol, Tom Gur, Subhayan Roy Moulik, Justin Thaler #### Quantum Proofs of Proximity We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property$\Pi$and reject ... more >>> TR21-067 | 6th May 2021 Zeyu Guo #### Variety Evasive Subspace Families Revisions: 1 We introduce the problem of constructing explicit variety evasive subspace families. Given a family$\mathcal{F}$of subvarieties of a projective or affine space, a collection$\mathcal{H}$of projective or affine$k$-subspaces is$(\mathcal{F},\epsilon)$-evasive if for every$\mathcal{V}\in\mathcal{F}$, all but at most$\epsilon$-fraction of$W\in\mathcal{H}$intersect every irreducible component of$\mathcal{V}$... more >>> TR21-066 | 5th May 2021 Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami #### Dimension-free Bounds and Structural Results in Communication Complexity The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to ... more >>> TR21-065 | 5th May 2021 Nikhil Mande, Swagato Sanyal #### One-way communication complexity and non-adaptive decision trees Revisions: 1 We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let$IP$denote Inner Product on ... more >>> TR21-064 | 5th May 2021 Noah Singer, Madhu Sudan, Santhoshini Velusamy #### Streaming approximation resistance of every ordering CSP Revisions: 2 An ordering constraint satisfaction problem (OCSP) is given by a positive integer$k$and a constraint predicate$\Pi$mapping permutations on$\{1,\ldots,k\}$to$\{0,1\}$. Given an instance of OCSP$(\Pi)$on$n$variables and$m$constraints, the goal is to find an ordering of the$n$variables that maximizes the number ... more >>> TR21-063 | 3rd May 2021 Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy #### Approximability of all finite CSPs in the dynamic streaming setting Revisions: 2 A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$for positive integers$q$and$k$. An instance of the problem on$n$variables is given by$m$applications of constraints from${\cal F}$to subsequences of the$n$... more >>> TR21-062 | 29th April 2021 Vishwas Bhargava, Sumanta Ghosh #### Improved Hitting Set for Orbit of ROABPs Revisions: 2 The orbit of an$n$-variate polynomial$f(\mathbf{x})$over a field$\mathbb{F}$is the set$\{f(A \mathbf{x} + b)\,\mid\, A\in \mathrm{GL}({n,\mathbb{F}})\mbox{ and }\mathbf{b} \in \mathbb{F}^n\}$, and the orbit of a polynomial class is the union of orbits of all the polynomials in it. In this paper, we give improved constructions of ... more >>> TR21-061 | 29th April 2021 Noah Fleming, Toniann Pitassi #### Reflections on Proof Complexity and Counting Principles This paper surveys the development of propositional proof complexity and the seminal contributions of Alasdair Urquhart. We focus on the central role of counting principles, and in particular Tseitin's graph tautologies, to most of the key advances in lower bounds in proof complexity. We reflect on a couple of key ... more >>> TR21-060 | 8th April 2021 Klim Efremenko, Gillat Kol, Raghuvansh Saxena #### Optimal Error Resilience of Adaptive Message Exchange We study the error resilience of the message exchange task: Two parties, each holding a private input, want to exchange their inputs. However, the channel connecting them is governed by an adversary that may corrupt a constant fraction of the transmissions. What is the maximum fraction of corruptions that still ... more >>> TR21-059 | 20th April 2021 Yanyi Liu, Rafael Pass #### On One-way Functions from NP-Complete Problems Revisions: 1 We present the first natural$\NP$-complete problem whose average-case hardness w.r.t. the uniform distribution over instances implies the existence of one-way functions (OWF). In fact, we prove that the existence of OWFs is \emph{equivalent} to mild average-case hardness of this$\NP$-complete problem. The problem, which originated in the 1960s, is ... more >>> TR21-058 | 21st April 2021 Shuichi Hirahara #### Average-Case Hardness of NP from Exponential Worst-Case Hardness Assumptions A long-standing and central open question in the theory of average-case complexity is to base average-case hardness of NP on worst-case hardness of NP. A frontier question along this line is to prove that PH is hard on average if UP requires (sub-)exponential worst-case complexity. The difficulty of resolving this ... more >>> TR21-057 | 23rd April 2021 Hanlin Ren, Rahul Santhanam #### Hardness of KT Characterizes Parallel Cryptography Revisions: 1 A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity K^t is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures: 1. We show, perhaps surprisingly, that the ... more >>> TR21-056 | 22nd April 2021 Yanyi Liu, Rafael Pass #### On the Possibility of Basing Cryptography on$\EXP \neq \BPP$Liu and Pass (FOCS'20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity---namely, Levin's notion of Kolmogorov Complexity---whose hardness is closely ... more >>> TR21-055 | 20th April 2021 Yanyi Liu, Rafael Pass #### Cryptography from Sublinear-Time Average-Case Hardness of Time-Bounded Kolmogorov Complexity Let$\mktp[s]$be the set of strings$x$such that$K^t(x) \leq s(|x|)$, where$K^t(x)$denotes the$t$-bounded Kolmogorov complexity of the truthtable described by$x$. Our main theorem shows that for an appropriate notion of mild average-case hardness, for every$\varepsilon>0$, polynomial$t(n) \geq (1+\varepsilon)n$, and every nice'' class ... more >>> TR21-054 | 14th April 2021 James Cook, Ian Mertz #### Encodings and the Tree Evaluation Problem We show that the Tree Evaluation Problem with alphabet size$k$and height$h$can be solved by branching programs of size$k^{O(h/\log h)} + 2^{O(h)}$. This answers a longstanding challenge of Cook et al. (2009) and gives the first general upper bound since the problem's inception. more >>> TR21-053 | 13th April 2021 Jan Krajicek #### Information in propositional proofs and algorithmic proof search We study from the proof complexity perspective the (informal) proof search problem: Is there an optimal way to search for propositional proofs? We note that for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal ... more >>> TR21-052 | 12th April 2021 Benny Applebaum, Oded Nir #### Upslices, Downslices, and Secret-Sharing with Complexity of$1.5^n$A secret-sharing scheme allows to distribute a secret$s$among$n$parties such that only some predefined authorized'' sets of parties can reconstruct the secret, and all other unauthorized'' sets learn nothing about$s$. The collection of authorized/unauthorized sets can be captured by a monotone function$f:\{0,1\}^n\rightarrow \{0,1\}$. more >>> TR21-051 | 8th April 2021 Klim Efremenko, Gillat Kol, Raghuvansh Saxena #### Binary Interactive Error Resilience Beyond$1/8$(or why$(1/2)^3 > 1/8$) Interactive error correcting codes are codes that encode a two party communication protocol to an error-resilient protocol that succeeds even if a constant fraction of the communicated symbols are adversarially corrupted, at the cost of increasing the communication by a constant factor. What is the largest fraction of corruptions that ... more >>> TR21-050 | 2nd April 2021 Marshall Ball, Alper Cakan, Tal Malkin #### Linear Threshold Secret-Sharing with Binary Reconstruction Motivated in part by applications in lattice-based cryptography, we initiate the study of the size of linear threshold ($t$-out-of-$n$') secret-sharing where the linear reconstruction function is restricted to coefficients in$\{0,1\}$. We prove upper and lower bounds on the share size of such schemes. One ramification of our results is ... more >>> TR21-049 | 1st April 2021 Juraj Hromkovic #### Kolmogorov complexity and nondeterminism versus determinism for polynomial time computations Revisions: 1 We call any consistent and sufficiently powerful formal theory that enables to algorithmically in polynomial time verify whether a text is a proof \textbf{efficiently verifiable mathematics} (ev-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of ev-mathematics. Our main results ... more >>> TR21-048 | 27th March 2021 William Hoza #### Better Pseudodistributions and Derandomization for Space-Bounded Computation Revisions: 1 Three decades ago, Nisan constructed an explicit pseudorandom generator (PRG) that fools width-$n$length-$n$read-once branching programs (ROBPs) with error$\varepsilon$and seed length$O(\log^2 n + \log n \cdot \log(1/\varepsilon))$(Combinatorica 1992). Nisan's generator remains the best explicit PRG known for this important model of computation. However, a recent ... more >>> TR21-047 | 26th March 2021 Zander Kelley, Raghu Meka #### Random restrictions and PRGs for PTFs in Gaussian Space A polynomial threshold function (PTF)$f:\mathbb{R}^n \rightarrow \mathbb{R}$is a function of the form$f(x) = sign(p(x))$where$p$is a polynomial of degree at most$d$. PTFs are a classical and well-studied complexity class with applications across complexity theory, learning theory, approximation theory, quantum complexity and more. We address ... more >>> TR21-046 | 22nd March 2021 Uma Girish, Avishay Tal, Kewen Wu #### Fourier Growth of Parity Decision Trees We prove that for every parity decision tree of depth$d$on$n$variables, the sum of absolute values of Fourier coefficients at level$\ell$is at most$d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$. Our result is nearly tight for small values of$\ell$and extends a previous Fourier bound ... more >>> TR21-045 | 22nd March 2021 Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich #### Reconstruction Algorithms for Low-Rank Tensors and Depth-3 Multilinear Circuits We give new and efficient black-box reconstruction algorithms for some classes of depth-$3$arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a {\it constant-rank} tensor. ... more >>> TR21-044 | 14th February 2021 Alexander Kulikov, Nikita Slezkin #### SAT-based Circuit Local Improvement Finding exact circuit size is a notorious optimization problem in practice. Whereas modern computers and algorithmic techniques allow to find a circuit of size seven in blink of an eye, it may take more than a week to search for a circuit of size thirteen. One of the reasons of ... more >>> TR21-043 | 15th March 2021 Peter Dixon, A. Pavan, N. V. Vinodchandran #### Promise Problems Meet Pseudodeterminism The Acceptance Probability Estimation Problem (APEP) is to additively approximate the acceptance probability of a Boolean circuit. This problem admits a probabilistic approximation scheme. A central question is whether we can design a pseudodeterministic approximation algorithm for this problem: a probabilistic polynomial-time algorithm that outputs a canonical approximation with high ... more >>> TR21-042 | 16th March 2021 Dana Moshkovitz #### Strong Parallel Repetition for Unique Games on Small Set Expanders Revisions: 1 We show that NP-hardness of approximating Boolean unique games on small set expanders can be amplified to the full Unique Games Conjecture on small set expanders. The latter conjecture is known to imply hardness results for problems like Balanced-Separator, Minimum-Linear-Rearrangement and Small-Set-Expansion that are not known under the Unique ... more >>> TR21-041 | 15th March 2021 Zhenjian Lu, Igor Carboni Oliveira #### An Efficient Coding Theorem via Probabilistic Representations and its Applications A probabilistic representation of a string$x \in \{0,1\}^n$is given by the code of a randomized algorithm that outputs$x$with high probability [Oliveira, ICALP 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ... more >>> TR21-040 | 15th March 2021 Lijie Chen, Zhenjian Lu, Xin Lyu, Igor Carboni Oliveira #### Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC1 We develop a general framework that characterizes strong average-case lower bounds against circuit classes$\mathcal{C}$contained in$\mathrm{NC}^1$, such as$\mathrm{AC}^0[\oplus]$and$\mathrm{ACC}^0$. We apply this framework to show: - Generic seed reduction: Pseudorandom generators (PRGs) against$\mathcal{C}$of seed length$\leq n -1$and error$\varepsilon(n) = n^{-\omega(1)}$can ... more >>> TR21-039 | 15th March 2021 Zhenjian Lu, Igor Carboni Oliveira, Rahul Santhanam #### Pseudodeterministic Algorithms and the Structure of Probabilistic Time We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of$BPTIME$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows. 1. A new pseudorandom generator and its consequences: We build on techniques developed to ... more >>> TR21-038 | 15th March 2021 Alessandro Chiesa, Fermi Ma, Nicholas Spooner, Mark Zhandry #### Post-Quantum Succinct Arguments Revisions: 1 We prove that Kilian's four-message succinct argument system is post-quantum secure in the standard model when instantiated with any probabilistically checkable proof and any collapsing hash function (which in turn exist based on the post-quantum hardness of Learning with Errors). At the heart of our proof is a new ... more >>> TR21-037 | 1st March 2021 Prerona Chatterjee #### Separating ABPs and Some Structured Formulas in the Non-Commutative Setting The motivating question for this work is a long standing open problem, posed by Nisan (1991), regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question continues to remain open, we make some progress towards its resolution. To that effect, ... more >>> TR21-036 | 14th March 2021 Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Madhu Sudan #### Ideal-theoretic Explanation of Capacity-achieving Decoding In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their ... more >>> TR21-035 | 13th March 2021 Robert Robere, Jeroen Zuiddam #### Amortized Circuit Complexity, Formal Complexity Measures, and Catalytic Algorithms We study the amortized circuit complexity of boolean functions. Given a circuit model$\mathcal{F}$and a boolean function$f : \{0,1\}^n \rightarrow \{0,1\}$, the$\mathcal{F}$-amortized circuit complexity is defined to be the size of the smallest circuit that outputs$m$copies of$f$(evaluated on the same input), ... more >>> TR21-034 | 9th March 2021 Oded Goldreich #### Robust Self-Ordering versus Local Self-Ordering We study two notions that refers to asymmetric graphs, which we view as graphs having a unique ordering that can be reconstructed by looking at an unlabeled version of the graph. A {\em local self-ordering} procedure for a graph$G$is given oracle access to an arbitrary isomorphic copy of ... more >>> TR21-033 | 7th March 2021 Susanna de Rezende #### Automating Tree-Like Resolution in Time$n^{o(\log n)}$Is ETH-Hard We show that tree-like resolution is not automatable in time$n^{o(\log n)}$unless ETH is false. This implies that, under ETH, the algorithm given by Beame and Pitassi (FOCS 1996) that automates tree-like resolution in time$n^{O(\log n)}$is optimal. We also provide a simpler proof of the result of ... more >>> TR21-032 | 5th March 2021 Justin Holmgren, Alex Lombardi, Ron Rothblum #### Fiat-Shamir via List-Recoverable Codes (or: Parallel Repetition of GMW is not Zero-Knowledge) Shortly after the introduction of zero-knowledge proofs, Goldreich, Micali and Wigderson (CRYPTO '86) demonstrated their wide applicability by constructing zero-knowledge proofs for the NP-complete problem of graph 3-coloring. A long-standing open question has been whether parallel repetition of their protocol preserves zero knowledge. In this work, we answer this question ... more >>> TR21-031 | 3rd March 2021 Vaibhav Krishan #### Upper Bound for Torus Polynomials We prove that all functions that have low degree torus polynomials approximating them with small error also have$MidBit^+$circuits computing them. This serves as a partial converse to the result that all$ACC$functions have low degree torus polynomials approximating them with small error, by Bhrushundi, Hosseini, Lovett and ... more >>> TR21-030 | 2nd March 2021 Shuichi Hirahara, Rahul Ilango, Bruno Loff #### Hardness of Constant-round Communication Complexity How difficult is it to compute the communication complexity of a two-argument total Boolean function$f:[N]\times[N]\to\{0,1\}$, when it is given as an$N\times N$binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard. In this ... more >>> TR21-029 | 1st March 2021 Inbar Kaslasi, Ron Rothblum, Prashant Nalini Vasudevan #### Public-Coin Statistical Zero-Knowledge Batch Verification against Malicious Verifiers Suppose that a problem$\Pi$has a statistical zero-knowledge (SZK) proof with communication complexity$m$. The question of batch verification for SZK asks whether one can prove that$k$instances$x_1,\ldots,x_k$all belong to$\Pi$with a statistical zero-knowledge proof whose communication complexity is better than$k \cdot m$(which ... more >>> TR21-028 | 27th February 2021 Anastasia Sofronova, Dmitry Sokolov #### Branching Programs with Bounded Repetitions and$\mathrm{Flow}$Formulas Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. '95 who showed the equivalence between regular Resolution and read-once branching programs ... more >>> TR21-027 | 24th February 2021 Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena, Zhao Song, Huacheng Yu #### Almost Optimal Super-Constant-Pass Streaming Lower Bounds for Reachability We give an almost quadratic$n^{2-o(1)}$lower bound on the space consumption of any$o(\sqrt{\log n})$-pass streaming algorithm solving the (directed)$s$-$t$reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including ... more >>> TR21-026 | 23rd February 2021 Joshua Brakensiek, Venkatesan Guruswami, Sai Sandeep #### Conditional Dichotomy of Boolean Ordered Promise CSPs Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their ... more >>> TR21-025 | 15th February 2021 Sivakanth Gopi, Venkatesan Guruswami #### Improved Maximally Recoverable LRCs using Skew Polynomials An$(n,r,h,a,q)$-Local Reconstruction Code is a linear code over$\mathbb{F}_q$of length$n$, whose codeword symbols are partitioned into$n/r$local groups each of size$r$. Each local group satisfies $a$' local parity checks to recover from $a$' erasures in that local group and there are further$h$global parity ... more >>> TR21-024 | 15th February 2021 Mika Göös, Gilbert Maystre #### A Majority Lemma for Randomised Query Complexity We show that computing the majority of$n$copies of a boolean function$g$has randomised query complexity$\mathrm{R}(\mathrm{Maj} \circ g^n) = \Theta(n\cdot \bar{\mathrm{R}}_{1/n}(g))$. In fact, we show that to obtain a similar result for any composed function$f\circ g^n$, it suffices to prove a sufficiently strong form of the ... more >>> TR21-023 | 20th February 2021 Jiatu Li, Tianqi Yang ####$3.1n - o(n)$Circuit Lower Bounds for Explicit Functions Proving circuit lower bounds has been an important but extremely hard problem for decades. Although one may show that almost every function$f:\mathbb{F}_2^n\to\mathbb{F}_2$requires circuit of size$\Omega(2^n/n)$by a simple counting argument, it remains unknown whether there is an explicit function (for example, a function in$NP$) not computable ... more >>> TR21-022 | 20th February 2021 Stefan Dantchev, Nicola Galesi, Abdul Ghani, Barnaby Martin #### Depth lower bounds in Stabbing Planes for combinatorial principles We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in ... more >>> TR21-021 | 18th February 2021 Per Austrin, Kilian Risse #### Average-Case Perfect Matching Lower Bounds from Hardness of Tseitin Formulas We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree$\Omega(n/\log n)$in the Polynomial ... more >>> TR21-020 | 15th February 2021 Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, Amnon Ta-Shma #### Error Reduction For Weighted PRGs Against Read Once Branching Programs Weighted pseudorandom generators (WPRGs), introduced by Braverman, Cohen and Garg [BCG20], is a generalization of pseudorandom generators (PRGs) in which arbitrary real weights are considered rather than a probability mass. Braverman et al. constructed WPRGs against read once branching programs (ROBPs) with near-optimal dependence on the error parameter. Chattopadhyay and ... more >>> TR21-019 | 17th February 2021 Edward Pyne, Salil Vadhan #### Pseudodistributions That Beat All Pseudorandom Generators Revisions: 1 A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ... more >>> TR21-018 | 20th February 2021 Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, Salil Vadhan #### Monotone Branching Programs: Pseudorandomness and Circuit Complexity Revisions: 1 We study monotone branching programs, wherein the states at each time step can be ordered so that edges with the same labels never cross each other. Equivalently, for each fixed input, the transition functions are a monotone function of the state. We prove that constant-width monotone branching programs of ... more >>> TR21-017 | 19th February 2021 Timothy Gowers, Emanuele Viola #### Mixing in non-quasirandom groups We initiate a systematic study of mixing in non-quasirandom groups. Let$A$and$B$be two independent, high-entropy distributions over a group$G$. We show that the product distribution$AB$is statistically close to the distribution$F(AB)$for several choices of$G$and$F$, including: (1)$G$is the affine ... more >>> TR21-016 | 16th February 2021 Shalev Ben-David, Mika Göös, Siddhartha Jain, Robin Kothari #### Unambiguous DNFs from Hex Revisions: 1 We exhibit an unambiguous$k$-DNF formula that requires CNF width$\tilde{\Omega}(k^{1.5})$. Our construction is inspired by the board game Hex and it is vastly simpler than previous ones, which achieved at best an exponent of$1.22$. Our result is known to imply several other improved separations in query and communication ... more >>> TR21-015 | 15th February 2021 Chandan Saha, Bhargav Thankey #### Hitting Sets for Orbits of Circuit Classes and Polynomial Families Revisions: 2 The orbit of an$n$-variate polynomial$f(\mathbf{x})$over a field$\mathbb{F}$is the set$\mathrm{orb}(f) := \{f(A\mathbf{x}+\mathbf{b}) : A \in \mathrm{GL}(n,\mathbb{F}) \ \mathrm{and} \ \mathbf{b} \in \mathbb{F}^n\}$. This paper studies explicit hitting sets for the orbits of polynomials computable by certain well-studied circuit classes. This version of the hitting set ... more >>> TR21-014 | 15th February 2021 Dori Medini, Amir Shpilka #### Hitting Sets and Reconstruction for Dense Orbits in VP$_e$and$\Sigma\Pi\Sigma$Circuits In this paper we study polynomials in VP$_e$(polynomial-sized formulas) and in$\Sigma\Pi\Sigma$(polynomial-size depth-$3$circuits) whose orbits, under the action of the affine group GL$^{aff}_n({\mathbb F})$, are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. As ... more >>> TR21-013 | 20th January 2021 Srinivasan Arunachalam, Penghui Yao #### Positive spectrahedrons: Geometric properties, Invariance principles and Pseudorandom generators In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary$m$-facet polytopes in$n$variables with seed length poly-logarithmic in$m,n$, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, ... more >>> TR21-012 | 9th February 2021 Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, Avi Wigderson #### On the Power and Limitations of Branch and Cut Revisions: 1 The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical ... more >>> TR21-011 | 13th February 2021 Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy #### Classification of the streaming approximability of Boolean CSPs Revisions: 3 , Comments: 1 A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint$f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of$m$constraint applications on$n$Boolean variables, where each constraint application applies the constraint to$k$literals chosen from the$n$variables and their negations. The goal ... more >>> TR21-010 | 11th February 2021 Eric Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle #### Cryptographic Hardness under Projections for Time-Bounded Kolmogorov Complexity A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT ... more >>> TR21-009 | 1st February 2021 Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, Ilya Volkovich #### One-way Functions and Partial MCSP Revisions: 1 , Comments: 1 One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence ... more >>> TR21-008 | 30th January 2021 Akash Kumar, C. Seshadhri, Andrew Stolman #### Random walks and forbidden minors III: poly(d/?)-time partition oracles for minor-free graph classes Revisions: 3 Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and n be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, for a sufficiently small ? > 0, one removes ?dn ... more >>> TR21-007 | 14th January 2021 Sai Sandeep #### Almost Optimal Inapproximability of Multidimensional Packing Problems Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be$d$-dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. ... more >>> TR21-006 | 18th January 2021 Susanna de Rezende, Jakob Nordström, Marc Vinyals #### How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity) We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large ... more >>> TR21-005 | 13th January 2021 Anindya De, Elchanan Mossel, Joe Neeman #### Robust testing of low-dimensional functions A natural problem in high-dimensional inference is to decide if a classifier$f:\mathbb{R}^n \rightarrow \{-1,1\}$depends on a small number of linear directions of its input data. Call a function$g: \mathbb{R}^n \rightarrow \{-1,1\}$, a linear$k$-junta if it is completely determined by some$k$-dimensional subspace of the input space. ... more >>> TR21-004 | 10th January 2021 Vishnu Iyer, Avishay Tal, Michael Whitmeyer #### Junta Distance Approximation with Sub-Exponential Queries Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function$f:\{\pm1\}^{n} \to \{\pm1\}$we give a poly$(k, \frac{1}{\varepsilon})$query algorithm that distinguishes between functions that are$\gamma$-close to$k$-juntas and$(\gamma+\varepsilon)$-far from ... more >>> TR21-003 | 6th January 2021 Lijie Chen, Xin Lyu #### Inverse-Exponential Correlation Bounds and Extremely Rigid Matrices from a New Derandomized XOR Lemma In this work we prove that there is a function$f \in \textrm{E}^\textrm{NP}$such that, for every sufficiently large$n$and$d = \sqrt{n}/\log n$,$f_n$($f$restricted to$n$-bit inputs) cannot be$(1/2 + 2^{-d})$-approximated by$\textrm{F}_2$-polynomials of degree$d$. We also observe that a minor improvement ... more >>> TR21-002 | 8th January 2021 Pooya Hatami, William Hoza, Avishay Tal, Roei Tell #### Fooling Constant-Depth Threshold Circuits Revisions: 1 We present new constructions of pseudorandom generators (PRGs) for two of the most widely-studied non-uniform circuit classes in complexity theory. Our main result is a construction of the first non-trivial PRG for linear threshold (LTF) circuits of arbitrary constant depth and super-linear size. This PRG fools circuits with depth$d\in\mathbb{N}$... more >>> TR21-001 | 1st January 2021 Klim Efremenko, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena #### Computation Over the Noisy Broadcast Channel with Malicious Parties We study the$n\$-party noisy broadcast channel with a constant fraction of malicious parties. Specifically, we assume that each non-malicious party holds an input bit, and communicates with the others in order to learn the input bits of all non-malicious parties. In each communication round, one of the parties broadcasts ... more >>>

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