Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > 2020:
All reports in year 2020:
TR20-082 | 23rd May 2020
Yuval Filmus, Meena Mahajan, Gaurav Sood, Marc Vinyals

#### MaxSAT Resolution and Subcube Sums

We study the MaxRes rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system MaxResW), p-simulates tree-like resolution. In devising a lower bound technique specific to MaxRes (and not merely inheriting lower bounds from ... more >>>

TR20-081 | 21st May 2020
Robert Andrews

#### Algebraic Hardness versus Randomness in Low Characteristic

We show that lower bounds for explicit constant-variate polynomials over fields of characteristic $p > 0$ are sufficient to derandomize polynomial identity testing over fields of characteristic $p$. In this setting, existing work on hardness-randomness tradeoffs for polynomial identity testing requires either the characteristic to be sufficiently large or the ... more >>>

TR20-080 | 19th May 2020
Joan Bruna, Oded Regev, Min Jae Song, Yi Tang

#### Continuous LWE

We introduce a continuous analogue of the Learning with Errors (LWE) problem, which we name CLWE. We give a polynomial-time quantum reduction from worst-case lattice problems to CLWE, showing that CLWE enjoys similar hardness guarantees to those of LWE. Alternatively, our result can also be seen as opening new avenues ... more >>>

TR20-079 | 15th May 2020
Hermann Gruber , Markus Holzer, Simon Wolfsteiner

#### On Minimizing Regular Expressions Without Kleene Star

Finite languages lie at the heart of literally every regular expression. Therefore, we investigate the approximation complexity of minimizing regular expressions without Kleene star, or, equivalently, regular expressions describing finite languages. On the side of approximation hardness, given such an expression of size~$s$, we prove that it is impossible to ... more >>>

TR20-078 | 21st May 2020
Eric Allender

#### The New Complexity Landscape around Circuit Minimization

We survey recent developments related to the Minimum Circuit Size Problem

more >>>

TR20-077 | 19th May 2020
Amit Sinhababu, Thomas Thierauf

#### Factorization of Polynomials Given by Arithmetic Branching Programs

Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size $s$, we show that all its factors can be computed by arithmetic branching programs of size $\text{poly}(s)$. Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was ... more >>>

TR20-076 | 17th May 2020
Benny Applebaum, Eliran Kachlon, Arpita Patra

#### The Round Complexity of Perfect MPC with Active Security and Optimal Resiliency

In STOC 1988, Ben-Or, Goldwasser, and Wigderson (BGW) established an important milestone in the fields of cryptography and distributed computing by showing that every functionality can be computed with perfect (information-theoretic and error-free) security at the presence of an active (aka Byzantine) rushing adversary that controls up to $n/3$ of ... more >>>

TR20-075 | 6th May 2020

#### Rigid Matrices From Rectangular PCPs

Revisions: 1

We introduce a variant of PCPs, that we refer to as *rectangular* PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the *row* of each query and the other determining the *column*.

We ... more >>>

TR20-074 | 6th May 2020
Eric Allender, Archit Chauhan, Samir Datta

#### Depth-First Search in Directed Graphs, Revisited

We present an algorithm for constructing a depth-first search tree in planar digraphs; the algorithm can be implemented in the complexity class UL, which is contained in nondeterministic logspace NL, which in turn lies in NC^2. Prior to this (for more than a quarter-century), the fastest uniform deterministic parallel algorithm ... more >>>

TR20-073 | 5th May 2020
Sam Buss, Dmitry Itsykson, Alexander Knop, Artur Riazanov, Dmitry Sokolov

#### Lower Bounds on OBDD Proofs with Several Orders

This paper is motivated by seeking lower bounds on OBDD($\land$, weakening, reordering) refutations, namely OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1-NBP($\land$) refutations based on read-once nondeterministic branching programs. These generalize OBDD($\land$, reordering) refutations. There are polynomial size 1-NBP($\land$) refutations of the pigeonhole principle, hence ... more >>>

TR20-072 | 5th May 2020
Yotam Dikstein, Irit Dinur, Prahladh Harsha, Noga Ron-Zewi

#### Locally testable codes via high-dimensional expanders

Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are far from all codewords, by probing a given word only at a very small (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. ... more >>>

TR20-071 | 4th May 2020
Iftach Haitner, Yonatan Karidi-Heller

#### A Tight Lower Bound on Adaptively Secure Full-Information Coin Flip

In a distributed coin-flipping protocol, Blum [ACM Transactions on Computer Systems '83],
the parties try to output a common (close to) uniform bit, even when some adversarially chosen parties try to bias the common output. In an adaptively secure full-information coin flip, Ben-Or and Linial [FOCS '85], the parties communicate ... more >>>

TR20-070 | 4th May 2020

#### On the list recoverability of randomly punctured codes

Revisions: 1

We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound.
In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound.
It was previously known that there are Reed-Solomon codes that do not have this ... more >>>

TR20-069 | 4th May 2020

#### Optimal Error Pseudodistributions for Read-Once Branching Programs

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$ and width $w$ read-once branching programs with seed length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error $\varepsilon$. It remains a central question to reduce the seed length to $O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$. However, there has ... more >>>

TR20-068 | 3rd May 2020
Oded Goldreich, Dana Ron

#### One-Sided Error Testing of Monomials and Affine Subspaces

Revisions: 1

We consider the query complexity of three versions of the problem of testing monomials and affine (and linear) subspaces with one-sided error, and obtain the following results:
\begin{itemize}
\item The general problem, in which the arity of the monomial (resp., co-dimension of the subspace) is not specified, has ... more >>>

TR20-067 | 30th April 2020
Dmitry Itsykson, Alexander Okhotin, Vsevolod Oparin

#### Computational and proof complexity of partial string avoidability

The partial string avoidability problem is stated as follows: given a finite set of strings with possible holes'' (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in ... more >>>

TR20-066 | 28th April 2020
Scott Aaronson, Shalev Ben-David, Robin Kothari, Avishay Tal

#### Quantum Implications of Huang's Sensitivity Theorem

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, the deterministic query complexity, $D(f)$, is at most quartic in the quantum query complexity, $Q(f)$: $D(f) = O(Q(f)^4)$. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, ... more >>>

TR20-065 | 2nd May 2020
Lijie Chen, Ce Jin, Ryan Williams

#### Sharp Threshold Results for Computational Complexity

We establish several sharp threshold'' results for computational complexity. For certain tasks, we can prove a resource lower bound of $n^c$ for $c \geq 1$ (or obtain an efficient circuit-analysis algorithm for $n^c$ size), there is strong intuition that a similar result can be proved for larger functions of $n$, ... more >>>

TR20-064 | 2nd May 2020
Mika Göös, Jakob Nordström, Toniann Pitassi, Robert Robere, Dmitry Sokolov, Susanna de Rezende

#### Automating Algebraic Proof Systems is NP-Hard

We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula $F$, it is NP-hard to find a refutation of $F$ in the Nullstellensatz, Polynomial Calculus, or Sherali--Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a ... more >>>

TR20-063 | 29th April 2020
Prerona Chatterjee, Mrinal Kumar, C Ramya, Ramprasad Saptharishi, Anamay Tengse

#### On the Existence of Algebraically Natural Proofs

For every constant c > 0, we show that there is a family {P_{N,c}} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, and that satisfies the following properties:
* For every family {f_n} of polynomials in VP, where f_n is an n ... more >>>

TR20-062 | 29th April 2020
Clement Canonne, Karl Wimmer

#### Testing Data Binnings

Motivated by the question of data quantization and "binning," we revisit the problem of identity testing of discrete probability distributions. Identity testing (a.k.a. one-sample testing), a fundamental and by now well-understood problem in distribution testing, asks, given a reference distribution (model) $\mathbf{q}$ and samples from an unknown distribution $\mathbf{p}$, both ... more >>>

TR20-061 | 28th April 2020
Deepanshu Kush, Benjamin Rossman

#### Tree-depth and the Formula Complexity of Subgraph Isomorphism

For a fixed "pattern" graph $G$, the $\textit{colored}$ $G\textit{-subgraph isomorphism problem}$ (denoted $\mathrm{SUB}(G)$) asks, given an $n$-vertex graph $H$ and a coloring $V(H) \to V(G)$, whether $H$ contains a properly colored copy of $G$. The complexity of this problem is tied to parameterized versions of $\mathit{P}$ ${=}?$ $\mathit{NP}$ and $\mathit{L}$ ... more >>>

TR20-060 | 23rd April 2020
Eshan Chattopadhyay, Jesse Goodman, Vipul Goyal, Xin Li

#### Leakage-Resilient Extractors and Secret-Sharing against Bounded Collusion Protocols

In a recent work, Kumar, Meka, and Sahai (FOCS 2019) introduced the notion of bounded collusion protocols (BCPs), in which $N$ parties wish to compute some joint function $f:(\{0,1\}^n)^N\to\{0,1\}$ using a public blackboard, but such that only $p$ parties may collude at a time. This generalizes well studied models in ... more >>>

TR20-059 | 16th April 2020
Gonen Krak, Noam Parzanchevski, Amnon Ta-Shma

#### Pr-ZSUBEXP is not contained in Pr-RP

We unconditionally prove there exists a promise problem in promise ZSUBEXP that cannot be solved in promise RP.
The proof technique builds upon Kabanets' easy witness method [Kab01] as implemented by Impagliazzo et. al [IKW02], with a separate diagonalization carried out on each of the two alternatives in the ... more >>>

TR20-058 | 24th April 2020
Shafi Goldwasser, Guy Rothblum, Jonathan Shafer, Amir Yehudayoff

#### Interactive Proofs for Verifying Machine Learning

We consider the following question: using a source of labeled data and interaction with an untrusted prover, what is the complexity of verifying that a given hypothesis is "approximately correct"? We study interactive proof systems for PAC verification, where a verifier that interacts with a prover is required to accept ... more >>>

TR20-057 | 20th April 2020
Alexander Golovnev, Gleb Posobin, Oded Regev, Omri Weinstein

#### Polynomial Data Structure Lower Bounds in the Group Model

Proving super-logarithmic data structure lower bounds in the static \emph{group model} has been a fundamental challenge in computational geometry since the early 80's. We prove a polynomial ($n^{\Omega(1)}$) lower bound for an explicit range counting problem of $n^3$ convex polygons in $\R^2$ (each with $n^{\tilde{O}(1)}$ facets/semialgebraic-complexity), against linear storage arithmetic ... more >>>

TR20-056 | 17th April 2020
James Cook, Ian Mertz

The study of branching programs for the Tree Evaluation Problem, introduced by S. Cook et al. (TOCT 2012), remains one of the most promising approaches to separating L from P. Given a label in $[k]$ at each leaf of a complete binary tree and an explicit function in $[k]^2 \to ... more >>> TR20-055 | 22nd April 2020 Ashutosh Kumar, Raghu Meka, David Zuckerman #### Bounded Collusion Protocols, Cylinder-Intersection Extractors and Leakage-Resilient Secret Sharing In this work we study bounded collusion protocols (BCPs) recently introduced in the context of secret sharing by Kumar, Meka, and Sahai (FOCS 2019). These are multi-party communication protocols on$n$parties where in each round a subset of$p$-parties (the collusion bound) collude together and write a function of ... more >>> TR20-054 | 22nd April 2020 Marshall Ball, Oded Goldreich, Tal Malkin #### Communication Complexity with Defective Randomness Starting with the two standard model of randomized communication complexity, we study the communication complexity of functions when the protocol has access to a defective source of randomness. Specifically, we consider both the public-randomness and private-randomness cases, while replacing the commonly postulated perfect randomness with distributions over$\ell$bit ... more >>> TR20-053 | 16th April 2020 Olaf Beyersdorff, Benjamin Böhm #### Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that ... more >>> TR20-052 | 14th April 2020 Yanyi Liu, Rafael Pass #### On One-way Functions and Kolmogorov Complexity Revisions: 1 We prove the equivalence of two fundamental problems in the theory of computation: - Existence of one-way functions: the existence of one-way functions (which in turn are equivalent to pseudorandom generators, pseudorandom functions, private-key encryption schemes, digital signatures, commitment schemes, and more). - Mild average-case hardness of$K^{poly}$-complexity: ... more >>> TR20-051 | 15th April 2020 Rafael Pass, Muthuramakrishnan Venkitasubramaniam #### Is it Easier to Prove Theorems that are Guaranteed to be True? Consider the following two fundamental open problems in complexity theory: (a) Does a hard-on-average language in$\NP$imply the existence of one-way functions?, or (b) Does a hard-on-average language in NP imply a hard-on-average problem in TFNP (i.e., the class of total NP search problem)? Our main result is that ... more >>> TR20-050 | 18th April 2020 Shuichi Hirahara #### Unexpected Hardness Results for Kolmogorov Complexity Under Uniform Reductions Hardness of computing the Kolmogorov complexity of a given string is closely tied to a security proof of hitting set generators, and thus understanding hardness of Kolmogorov complexity is one of the central questions in complexity theory. In this paper, we develop new proof techniques for showing hardness of computing ... more >>> TR20-049 | 17th April 2020 Mika Göös, Sajin Koroth, Ian Mertz, Toniann Pitassi #### Automating Cutting Planes is NP-Hard We show that Cutting Planes (CP) proofs are hard to find: Given an unsatisfiable formula$F$, (1) it is NP-hard to find a CP refutation of$F$in time polynomial in the length of the shortest such refutation; and (2) unless Gap-Hitting-Set admits a nontrivial algorithm, one cannot find a ... more >>> TR20-048 | 16th April 2020 Shachar Lovett, Raghu Meka, Jiapeng Zhang #### Improved lifting theorems via robust sunflowers Lifting theorems are a generic way to lift lower bounds in query complexity to lower bounds in communication complexity, with applications in diverse areas, such as combinatorial optimization, proof complexity, game theory. Lifting theorems rely on a gadget, where smaller gadgets give stronger lower bounds. However, existing proof techniques are ... more >>> TR20-047 | 16th April 2020 Ronen Shaltiel, Jad Silbak #### Explicit Uniquely Decodable Codes for Space Bounded Channels That Achieve List-Decoding Capacity We consider codes for space bounded channels. This is a model for communication under noise that was introduced by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in ... more >>> TR20-046 | 13th April 2020 Srikanth Srinivasan #### A Robust Version of Heged\H{u}s's Lemma, with Applications Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field$\mathbb{F}$of characteristic$p > 0$and for$q$a power of$p$, the lemma says that any multilinear polynomial$P\in \mathbb{F}[x_1,\ldots,x_n]$of degree less than$q$that vanishes at all points in$\{0,1\}^n$of ... more >>> TR20-045 | 15th April 2020 Ankit Garg, Neeraj Kayal, Chandan Saha #### Learning sums of powers of low-degree polynomials in the non-degenerate case We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an$n$-variate degree-$d$polynomial$f$which can be written as $$f = c_1Q_1^{m} + \ldots + c_s Q_s^{m},$$ where each$c_i\in \mathbb{F}^{\times}$,$Q_i$is a homogeneous polynomial of degree$t$, and$t m = ... more >>>

TR20-044 | 8th April 2020
Marshall Ball, Elette Boyle, Akshay Degwekar, Apoorvaa Deshpande, Alon Rosen, Vinod Vaikuntanathan, Prashant Vasudevan

#### Cryptography from Information Loss

Reductions between problems, the mainstay of theoretical computer science, efficiently map an instance of one problem to an instance of another in such a way that solving the latter allows solving the former. The subject of this work is lossy'' reductions, where the reduction loses some information about the input ... more >>>

TR20-043 | 29th March 2020
Dorit Aharonov, Alex Bredariol Grilo

#### A combinatorial MA-complete problem

Despite the interest in the complexity class MA, the randomized analog of NP, there is just a couple of known natural (promise-)MA-complete problems, the first due to Bravyi and Terhal (SIAM Journal of Computing 2009) and the second due to Bravyi (Quantum Information and Computation 2015). Surprisingly, both problems are ... more >>>

TR20-042 | 31st March 2020
Pranav Bisht, Nitin Saxena

#### Poly-time blackbox identity testing for sum of log-variate constant-width ROABPs

Blackbox polynomial identity testing (PIT) affords 'extreme variable-bootstrapping' (Agrawal et al, STOC'18; PNAS'19; Guo et al, FOCS'19). This motivates us to study log-variate read-once oblivious algebraic branching programs (ROABP). We restrict width of ROABP to a constant and study the more general sum-of-ROABPs model. We give the first poly($s$)-time blackbox ... more >>>

TR20-041 | 29th March 2020
Mrinal Kumar, Ben Lee Volk

#### A Polynomial Degree Bound on Defining Equations of Non-rigid Matrices and Small Linear Circuits

Revisions: 1

We show that there is a defining equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field $\mathbb{F}$, there is a non-zero $n^2$-variate polynomial $P \in \mathbb{F}(x_{1, 1}, \ldots, x_{n, n})$ of degree ... more >>>

TR20-040 | 25th March 2020
Andrei Krokhin, Jakub Opršal, Marcin Wrochna, Stanislav Zivny

#### Topology and adjunction in promise constraint satisfaction

The approximate graph colouring problem concerns colouring a $k$-colourable
graph with $c$ colours, where $c\geq k$. This problem naturally generalises
to promise graph homomorphism and further to promise constraint satisfaction
problems. Complexity analysis of all these problems is notoriously difficult.
In this paper, we introduce ... more >>>

TR20-039 | 25th March 2020
Pranjal Dutta, Nitin Saxena, Thomas Thierauf

#### Lower bounds on the sum of 25th-powers of univariates lead to complete derandomization of PIT

We consider the univariate polynomial $f_d:=(x+1)^d$ when represented as a sum of constant-powers of univariate polynomials. We define a natural measure for the model, the support-union, and conjecture that it is $\Omega(d)$ for $f_d$.

We show a stunning connection of the conjecture to the two main problems in algebraic ... more >>>

TR20-038 | 15th March 2020
Ofer Grossman, Justin Holmgren

#### Error Correcting Codes for Uncompressed Messages

Most types of messages we transmit (e.g., video, audio, images, text) are not fully compressed, since they do not have known efficient and information theoretically optimal compression algorithms. When transmitting such messages, standard error correcting codes fail to take advantage of the fact that messages are not fully compressed.

We ... more >>>

TR20-037 | 18th March 2020
Michal Garlik

#### Failure of Feasible Disjunction Property for $k$-DNF Resolution and NP-hardness of Automating It

We show that for every integer $k \geq 2$, the Res($k$) propositional proof system does not have the weak feasible disjunction property. Next, we generalize a recent result of Atserias and Müller [FOCS, 2019] to Res($k$). We show that if NP is not included in P (resp. QP, SUBEXP) then ... more >>>

TR20-036 | 9th March 2020
Olaf Beyersdorff, Joshua Blinkhorn, Tomáš Peitl

#### Strong (D)QBF Dependency Schemes via Tautology-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 tautology-free DQBF dependency scheme that generalises the reflexive resolution path dependency scheme. We establish soundness of the tautology-free scheme, implying that it can be used in any ... more >>>

TR20-035 | 23rd February 2020
Justin Holmgren

#### No-Signaling MIPs and Faster-Than-Light Communication, Revisited

We revisit one original motivation for the study of no-signaling multi-prover
interactive proofs (NS-MIPs): specifically, that two spatially separated
provers are guaranteed to be no-signaling by the physical principle that
information cannot travel from one point to another faster than light.

We observe that with ... more >>>

TR20-034 | 12th March 2020
Erfan Khaniki

#### On Proof complexity of Resolution over Polynomial Calculus

Revisions: 1

The refutation system ${Res}_R({PC}_d)$ is a natural extension of resolution refutation system such that it operates with disjunctions of degree $d$ polynomials over ring $R$ with boolean variables. For $d=1$, this system is called ${Res}_R({lin})$. Based on properties of $R$, ${Res}_R({lin})$ systems can be too strong to prove lower ... more >>>

TR20-033 | 12th March 2020
Suryajith Chillara

#### New Exponential Size Lower Bounds against Depth Four Circuits of Bounded Individual Degree

Revisions: 1

Kayal, Saha and Tavenas [Theory of Computing, 2018] showed that for all large enough integers $n$ and $d$ such that $d\geq \omega(\log{n})$, any syntactic depth four circuit of bounded individual degree $\delta = o(d)$ that computes the Iterated Matrix Multiplication polynomial ($IMM_{n,d}$) must have size $n^{\Omega\left(\sqrt{d/\delta}\right)}$. Unfortunately, this bound ... more >>>

TR20-032 | 12th March 2020
Suryajith Chillara

#### On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits

In this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree of $r$ (referred to as multi-$r$-ic circuits). The goal of this study is to make progress towards proving ... more >>>

TR20-031 | 10th March 2020
Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, Nitin Saurabh

#### Algebraic Branching Programs, Border Complexity, and Tangent Spaces

Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most $k$ is Zariski-closed, an important property in ... more >>>

TR20-030 | 9th March 2020
Matthias Christandl, François Le Gall, Vladimir Lysikov, Jeroen Zuiddam

#### Barriers for Rectangular Matrix Multiplication

We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give optimal algorithms. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously ... more >>>

TR20-029 | 6th March 2020
Swastik Kopparty, Guy Moshkovitz, Jeroen Zuiddam

#### Geometric Rank of Tensors and Subrank of Matrix Multiplication

Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the ... more >>>

TR20-028 | 27th February 2020
Nikhil Gupta, Chandan Saha, Bhargav Thankey

#### A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits

We show an $\widetilde{\Omega}(n^{2.5})$ lower bound for general depth four arithmetic circuits computing an explicit $n$-variate degree $\Theta(n)$ multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey by Shpilka and Yehudayoff (FnT-TCS, 2010), no super-quadratic lower bound was known for depth four ... more >>>

TR20-027 | 26th February 2020
Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, Li-Yang Tan

#### The Power of Many Samples in Query Complexity

The randomized query complexity $R(f)$ of a boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is famously characterized (via Yao's minimax) by the least number of queries needed to distinguish a distribution $D_0$ over $0$-inputs from a distribution $D_1$ over $1$-inputs, maximized over all pairs $(D_0,D_1)$. We ask: Does this task become easier if we ... more >>>

TR20-026 | 25th February 2020
Dean Doron, Jack Murtagh, Salil Vadhan, David Zuckerman

Revisions: 1

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\leq \log n$ and an error parameter $\varepsilon > 0$, our algorithm runs in space $\tilde{O}(k\log (N\cdot ... more >>> TR20-025 | 20th February 2020 Chetan Gupta, Vimal Raj Sharma, Raghunath Tewari #### Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs We show that given an embedding of an O(log n) genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists. As a consequence, we obtain that deciding whether the ... more >>> TR20-024 | 20th February 2020 Samir Datta, Chetan Gupta, Rahul Jain, Vimal Raj Sharma, Raghunath Tewari #### Randomized and Symmetric Catalytic Computation A catalytic Turing machine is a model of computation that is created by equipping a Turing machine with an additional auxiliary tape which is initially filled with arbitrary content; the machine can read or write on auxiliary tape during the computation but when it halts auxiliary tape’s initial content must ... more >>> TR20-023 | 10th February 2020 Marshall Ball, Eshan Chattopadhyay, Jyun-Jie Liao, Tal Malkin, Li-Yang Tan #### Non-Malleability against Polynomial Tampering We present the first explicit construction of a non-malleable code that can handle tampering functions that are bounded-degree polynomials. Prior to our work, this was only known for degree-1 polynomials (affine tampering functions), due to Chattopadhyay and Li (STOC 2017). As a direct corollary, we obtain an explicit non-malleable ... more >>> TR20-022 | 19th February 2020 Klim Efremenko, Gillat Kol, Raghuvansh Saxena #### Interactive Error Resilience Beyond$\frac{2}{7}$Interactive error correcting codes can protect interactive communication protocols against a constant fraction of adversarial errors, while incurring only a constant multiplicative overhead in the total communication. What is the maximum fraction of errors that such codes can protect against? For the non-adaptive channel, where the parties must agree ... more >>> TR20-021 | 21st February 2020 Rahul Ilango, Bruno Loff, Igor Carboni Oliveira #### NP-Hardness of Circuit Minimization for Multi-Output Functions Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an ... more >>> TR20-020 | 21st February 2020 Nikhil Mande, Justin Thaler, Shuchen Zhu #### Improved Approximate Degree Bounds For$k$-distinctness An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the$k$-distinctness function on inputs of size$N$. While the case of$k=2$(also called Element Distinctness) is well-understood, there is a polynomial gap between ... more >>> TR20-019 | 19th February 2020 Siddharth Bhandari, Prahladh Harsha #### A note on the explicit constructions of tree codes over polylogarithmic-sized alphabet Recently, Cohen, Haeupler and Schulman gave an explicit construction of binary tree codes over polylogarithmic-sized output alphabet based on Pudl\'{a}k's construction of maximum-distance-separable (MDS) tree codes using totally-non-singular triangular matrices. In this short note, we give a unified and simpler presentation of Pudl\'{a}k and Cohen-Haeupler-Schulman's constructions. more >>> TR20-018 | 18th February 2020 Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor Oliveira #### Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates The class$FORMULA[s] \circ \mathcal{G}$consists of Boolean functions computable by size-$s$de Morgan formulas whose leaves are any Boolean functions from a class$\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for$FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes$\mathcal{G}$of functions with low communication complexity. Let$R^{(k)}(\mathcal{G})$be ... more >>> TR20-017 | 18th February 2020 Alexander Kozachinskiy, Vladimir Podolskii #### Multiparty Karchmer-Wigderson Games and Threshold Circuits We suggest a generalization of Karchmer-Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone ... more >>> TR20-016 | 17th February 2020 Kuan Cheng, William Hoza #### Hitting Sets Give Two-Sided Derandomization of Small Space Revisions: 1 A hitting set is a "one-sided" variant of a pseudorandom generator (PRG), naturally suited to derandomizing algorithms that have one-sided error. We study the problem of using a given hitting set to derandomize algorithms that have two-sided error, focusing on space-bounded algorithms. For our first result, we show that if ... more >>> TR20-015 | 18th February 2020 Emanuele Viola #### New lower bounds for probabilistic degree and AC0 with parity gates Revisions: 2 We prove new lower bounds for computing some functions$f:\{0,1\}^{n}\to\{0,1\}$in$E^{NP}$by polynomials modulo$2$, constant-depth circuits with parity gates ($AC^{0}[\oplus]$), and related classes. Results include: (1)$\Omega(n/\log^{2}n)$lower bounds probabilistic degree. This is optimal up to a factor$O(\log^{2}n)$. The previous best lower bound was$\Omega(\sqrt{n})$proved in ... more >>> TR20-014 | 16th February 2020 Gil Cohen, Shahar Samocha #### Palette-Alternating Tree Codes A tree code is an edge-coloring of the complete infinite binary tree such that every two nodes of equal depth have a fraction--bounded away from$0$--of mismatched colors between the corresponding paths to their least common ancestor. Tree codes were introduced in a seminal work by Schulman (STOC 1993) and ... more >>> TR20-013 | 17th February 2020 Noga Ron-Zewi, Mary Wootters, Gilles Z\'{e}mor #### Linear-time Erasure List-decoding of Expander Codes We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let$r > 0$be any integer. Given an inner code$\cC_0$of length$d$, and a$d$-regular bipartite expander graph$G$with$n$vertices on each side, we give an algorithm to list-decode the expander code$\cC ... more >>>

TR20-012 | 14th February 2020
Dmitry Sokolov

#### (Semi)Algebraic Proofs over $\{\pm 1\}$ Variables

One of the major open problems in proof complexity is to prove lower bounds on $AC_0[p]$-Frege proof
systems. As a step toward this goal Impagliazzo, Mouli and Pitassi in a recent paper suggested to prove
lower bounds on the size for Polynomial Calculus over the $\{\pm 1\}$ basis. In this ... more >>>

TR20-011 | 9th February 2020
Dominik Scheder, Navid Talebanfard

#### Super Strong ETH is true for strong PPSZ

We construct $k$-CNFs with $m$ variables on which the strong version of PPSZ $k$-SAT algorithm, which uses bounded width resolution, has success probability at most $2^{-(1 - (1 + \epsilon)2/k)m}$ for every $\epsilon > 0$. Previously such a bound was known only for the weak PPSZ algorithm which exhaustively searches ... more >>>

TR20-010 | 12th February 2020
Lijie Chen, Hanlin Ren

#### Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization

We prove that for all constants a, NQP = NTIME[n^{polylog(n)}] cannot be (1/2 + 2^{-log^a n})-approximated by 2^{log^a n}-size ACC^0 of THR circuits (ACC^0 circuits with a bottom layer of THR gates). Previously, it was even open whether E^NP can be (1/2+1/sqrt{n})-approximated by AC^0[2] circuits. As a straightforward application, ... more >>>

TR20-009 | 6th February 2020
Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra

#### Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality

We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL
Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a
new approach: looking at the first Fourier level of the function after a suitable random restriction and
applying the Log-Sobolev ... more >>>

TR20-008 | 26th January 2020
Benny Applebaum, Amos Beimel, Oded Nir, Naty Peter

#### Better Secret-Sharing via Robust Conditional Disclosure of Secrets

Revisions: 1

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 sets is called the access structure. For over 30 years, it was ... more >>>

TR20-007 | 19th December 2019
Claude Crépeau, Arnaud Massenet, Louis Salvail, Lucas Stinchcombe, Nan Yang

#### Practical Relativistic Zero-Knowledge for NP

In this work we consider the following problem: in a Multi-Prover environment, how close can we get to prove the validity of an NP statement in Zero-Knowledge ? We exhibit a set of two novel Zero-Knowledge protocols for the 3-COLorability problem that use two (local) provers or three (entangled) provers ... more >>>

TR20-006 | 22nd January 2020
Anup Rao, Amir Yehudayoff

#### The Communication Complexity of the Exact Gap-Hamming Problem

We prove a sharp lower bound on the distributional communication complexity of the exact gap-hamming problem.

more >>>

TR20-005 | 17th January 2020
Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan

#### Hardness Characterisations and Size-Width Lower Bounds for QBF Resolution

Revisions: 1

We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) by circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems (Beyersdorff & Pich, LICS 2016), but leaving open the most important case of QBF resolution. Different from the Frege ... more >>>

TR20-004 | 17th January 2020
Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, Stanislav Zivny

#### The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not ... more >>>

TR20-003 | 15th January 2020
Giuseppe Persiano, Kevin Yeo

#### Tight Static Lower Bounds for Non-Adaptive Data Structures

Revisions: 1

In this paper, we study the static cell probe complexity of non-adaptive data structures that maintain a subset of $n$ points from a universe consisting of $m=n^{1+\Omega(1)}$ points. A data structure is defined to be non-adaptive when the memory locations that are chosen to be accessed during a query depend ... more >>>

TR20-002 | 6th January 2020
Sophie Laplante, Reza Naserasr, Anupa Sunny

#### Sensitivity lower bounds from linear dependencies

Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least the square root of n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we ... more >>>

TR20-001 | 31st December 2019
Or Meir, Jakob Nordström, Robert Robere, Susanna de Rezende

#### Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

Revisions: 2

We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph $G$ can be reversibly pebbled in time $t$ and space $s$ if and only if there is a Nullstellensatz refutation of the pebbling formula over $G$ in size $t+1$ ... more >>>

ISSN 1433-8092 | Imprint