Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > 2019:
All reports in year 2019:
TR19-088 | 16th June 2019
Oded Goldreich

#### On the Complexity of Estimating the Effective Support Size

Loosely speaking, the effective support size of a distribution is the size of the support of a distribution that is close to it (in totally variation distance).
We study the complexity of estimating the effective support size of an unknown distribution when given samples of the distributions as well ... more >>>

TR19-087 | 10th June 2019
Rohit Agrawal

In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in $\mathcal ... more >>> TR19-086 | 7th June 2019 Alex Bredariol Grilo, William Slofstra, Henry Yuen #### Perfect zero knowledge for quantum multiprover interactive proofs In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs — notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP$^*$, the set of languages decidable by multiprover interactive ... more >>> TR19-085 | 7th June 2019 Xuangui Huang, Emanuele Viola #### Approximate Degree-Weight and Indistinguishability We prove that the Or function on$n$bits can be point-wise approximated with error$\eps$by a polynomial of degree$O(k)$and weight$2^{O(n \log (1/\eps)/k)}$, for any$k \geq \sqrt{n \log 1/\eps}$. This result is tight for all$k$. Previous results were either not tight or had$\eps ... more >>>

TR19-084 | 26th May 2019
Michal Garlik

#### Resolution Lower Bounds for Refutation Statements

For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in [Atserias-Müller,2019] asks whether a certain natural propositional encoding of the above statement is ... more >>>

TR19-083 | 4th June 2019
Lior Gishboliner, Asaf Shapira

#### Testing Graphs against an Unknown Distribution

Revisions: 1

The area of graph property testing seeks to understand the relation between the global properties of a graph and its local statistics. In the classical model, the local statistics of a graph is defined relative to a uniform distribution over the graph’s vertex set. A graph property $\mathcal{P}$ is said ... more >>>

TR19-082 | 2nd June 2019
Andrej Bogdanov, Nikhil Mande, Justin Thaler, Christopher Williamson

#### Approximate degree, secret sharing, and concentration phenomena

The $\epsilon$-approximate degree $\widetilde{\text{deg}}_\epsilon(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $\epsilon$. The approximate degree of $f$ is at least $k$ iff there exists a pair of probability distributions, also known as a dual polynomial, that are perfectly ... more >>>

TR19-081 | 31st May 2019
Iftach Haitner, Noam Mazor, Ronen Shaltiel, Jad Silbak

#### Channels of Small Log-Ratio Leakage and Characterization of Two-Party Differentially Private Computation

Consider a PPT two-party protocol ?=(A,B) in which the parties get no private inputs and obtain outputs O^A,O^B?{0,1}, and let V^A and V^B denote the parties’ individual views. Protocol ? has ?-agreement if Pr[O^A=O^B]=1/2+?. The leakage of ? is the amount of information a party obtains about the event {O^A=O^B}; ... more >>>

TR19-080 | 1st June 2019
Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

#### On List Recovery of High-Rate Tensor Codes

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is {\em approximately} locally list recoverable, as well as globally list recoverable ... more >>>

TR19-079 | 28th May 2019
Arnab Bhattacharyya, Philips George John, Suprovat Ghoshal, Raghu Meka

#### Average Bias and Polynomial Sources

Revisions: 2

We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over $\{0,1\}^n$, its average bias is: $b_{\text{av}}(Z) =2^{-n} \sum_{c \in \{0,1\}^n} |\mathbb{E}_{z \sim Z}(-1)^{\langle c, z\rangle}|$. A source with average bias at most $2^{-k}$ has min-entropy at least $k$, and ... more >>>

TR19-078 | 1st June 2019
Itai Benjamini, Oded Goldreich

#### Pseudo-Mixing Time of Random Walks

We introduce the notion of pseudo-mixing time of a graph define as the number of steps in a random walk that suffices for generating a vertex that looks random to any polynomial-time observer, where, in addition to the tested vertex, the observer is also provided with oracle access to the ... more >>>

TR19-077 | 30th May 2019
Jan Bydzovsky, Igor Carboni Oliveira, Jan Krajicek

#### Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $P^{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere ... more >>>

TR19-076 | 24th May 2019
Leroy Chew, Judith Clymo

#### The Equivalences of Refutational QRAT

The solving of Quantified Boolean Formulas (QBF) has been advanced considerably in the last two decades. In response to this, several proof systems have been put forward to universally verify QBF solvers.
QRAT by Heule et al. is one such example of this and builds on technology from DRAT, ... more >>>

TR19-075 | 25th May 2019
Lijie Chen, Dylan McKay, Cody Murray, Ryan Williams

#### Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems

Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems

A frontier open problem in circuit complexity is to prove P^NP is not in SIZE[n^k] for all k; this is a necessary intermediate step towards NP is not in P/poly. Previously, for several classes containing P^NP, including NP^NP, ZPP^NP, and ... more >>>

TR19-074 | 22nd May 2019
Arka Rai Choudhuri, Pavel Hubacek, Chethan Kamath, Krzysztof Pietrzak, Alon Rosen, Guy Rothblum

#### Finding a Nash Equilibrium Is No Easier Than Breaking Fiat-Shamir

The Fiat-Shamir heuristic transforms a public-coin interactive proof into a non-interactive argument, by replacing the verifier with a cryptographic hash function that is applied to the protocol’s transcript. Constructing hash functions for which this transformation is sound is a central and long-standing open question in cryptography.

We show that ... more >>>

TR19-073 | 17th May 2019
Igor Carboni Oliveira, Rahul Santhanam, Srikanth Srinivasan

#### Parity helps to compute Majority

We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC$^0[\oplus]$ circuits. Razborov (1987) and Smolensky (1987, 1993) showed that Majority requires depth-$d$ AC$^0[\oplus]$ circuits of size $2^{\Omega(n^{1/2(d-1)})}$. By using a divide-and-conquer approach, it is easy to show that Majority can ... more >>>

TR19-072 | 17th May 2019
Lijie Chen, Ofer Grossman

#### Broadcast Congested Clique: Planted Cliques and Pseudorandom Generators

Consider the multiparty communication complexity model where there are n processors, each receiving as input a row of an n by n matrix M with entries in {0, 1}, and in each round each party can broadcast a single bit to all other parties (this is known as the BCAST(1) ... more >>>

TR19-071 | 14th May 2019
Sumegha Garg, Ran Raz, Avishay Tal

#### Time-Space Lower Bounds for Two-Pass Learning

A line of recent works showed that for a large class of learning problems, any learning algorithm requires either super-linear memory size or a super-polynomial number of samples [Raz16,KRT17,Raz17,MM18,BOGY18,GRT18]. For example, any algorithm for learning parities of size $n$ requires either a memory of size $\Omega(n^{2})$ or an exponential number ... more >>>

TR19-070 | 14th May 2019
Alessandro Chiesa, Peter Manohar, Igor Shinkar

#### On Local Testability in the Non-Signaling Setting

Non-signaling strategies are a generalization of quantum strategies that have been studied in physics for decades, and have recently found applications in theoretical computer science. These applications motivate the study of local-to-global phenomena for non-signaling functions.

We present general results about the local testability of linear codes in the non-signaling ... more >>>

TR19-069 | 6th May 2019
Nicola Galesi, Dmitry Itsykson, Artur Riazanov, Anastasia Sofronova

#### Bounded-depth Frege complexity of Tseitin formulas for all graphs

Revisions: 1

We prove that there is a constant $K$ such that \emph{Tseitin} formulas for an undirected graph $G$ requires proofs of
size $2^{\mathrm{tw}(G)^{\Omega(1/d)}}$ in depth-$d$ Frege systems for $d<\frac{K \log n}{\log \log n}$, where $\tw(G)$ is the treewidth of $G$. This extends H{\aa}stad recent lower bound for the grid graph ... more >>>

TR19-068 | 27th April 2019
Shuo Pang

#### LARGE CLIQUE IS HARD ON AVERAGE FOR RESOLUTION

We prove resolution lower bounds for $k$-Clique on the Erdos-Renyi random graph $G(n,n^{-{2\xi}\over{k-1}})$ (where $\xi>1$ is constant). First we show for $k=n^{c_0}$, $c_0\in(0,1/3)$, an $\exp({\Omega(n^{(1-\epsilon)c_0})})$ average lower bound on resolution where $\epsilon$ is arbitrary constant.

We then propose the model of $a$-irregular resolution. Extended from regular resolution, this model ... more >>>

TR19-067 | 6th May 2019
Hamed Hatami, Kaave Hosseini, Shachar Lovett

#### Sign rank vs Discrepancy

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions.
In this article, we establish the strongest possible separation by constructing a Boolean matrix whose sign-rank ... more >>>

TR19-066 | 3rd May 2019
Thomas Watson

#### A Lower Bound for Sampling Disjoint Sets

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set $x\subseteq[n]$ and Bob ends up with a set $y\subseteq[n]$, such that $(x,y)$ is uniformly distributed over all pairs of disjoint sets. ... more >>>

TR19-065 | 1st May 2019
Mrinal Kumar, Ramprasad Saptharishi, Noam Solomon

#### Derandomization from Algebraic Hardness: Treading the Borders

Revisions: 1

A hitting-set generator (HSG) is a polynomial map $Gen:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $Q$ of small enough circuit size and degree, if $Q$ is non-zero, then $Q\circ Gen$ is non-zero. In this paper, we give a new construction of such a HSG assuming that we have ... more >>>

TR19-064 | 23rd April 2019
Igor Carboni Oliveira

#### Randomness and Intractability in Kolmogorov Complexity

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin's notion of Kolmogorov complexity from 1984. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability.

This complexity measure gives rise to a ... more >>>

TR19-063 | 28th April 2019
Vikraman Arvind, Abhranil Chatterjee, Rajit Datta, Partha Mukhopadhyay

#### Efficient Black-Box Identity Testing for Free Group Algebra

Hrubeš and Wigderson [HW14] initiated the study of
noncommutative arithmetic circuits with division computing a
noncommutative rational function in the free skew field, and
raised the question of rational identity testing. It is now known
that the problem can be solved in deterministic polynomial time in
more >>>

TR19-062 | 18th April 2019
Scott Aaronson, Robin Kothari, William Kretschmer, Justin Thaler

#### Quantum Lower Bounds for Approximate Counting via Laurent Polynomials

This paper proves new limitations on the power of quantum computers to solve approximate counting---that is, multiplicatively estimating the size of a nonempty set $S\subseteq [N]$.

Given only a membership oracle for $S$, it is well known that approximate counting takes $\Theta(\sqrt{N/|S|})$ quantum queries. But what if a quantum algorithm ... more >>>

TR19-061 | 16th April 2019
Scott Aaronson, Daniel Grier, Luke Schaeffer

#### A Quantum Query Complexity Trichotomy for Regular Languages

We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity $\Theta(1)$, $\tilde{\Theta}(\sqrt n)$, or $\Theta(n)$. The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we ... more >>>

TR19-060 | 18th April 2019
Scott Aaronson, Guy Rothblum

#### Gentle Measurement of Quantum States and Differential Privacy

In differential privacy (DP), we want to query a database about $n$ users, in a way that "leaks at most $\varepsilon$ about any individual user," even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure $n$ quantum states, in a way that "damages the ... more >>>

TR19-059 | 18th April 2019
Rohit Agrawal

#### Samplers and extractors for unbounded functions

Blasiok (SODA'18) recently introduced the notion of a subgaussian sampler, defined as an averaging sampler for approximating the mean of functions $f:\{0,1\}^m \to \mathbb{R}$ such that $f(U_m)$ has subgaussian tails, and asked for explicit constructions. In this work, we give the first explicit constructions of subgaussian samplers (and in fact ... more >>>

TR19-058 | 16th April 2019
Pavel Pudlak, Vojtech Rodl

#### Extractors for small zero-fixing sources

A random variable $X$ is an $(n,k)$-zero-fixing source if for some subset $V\subseteq[n]$, $X$ is the uniform distribution on the strings $\{0,1\}^n$ that are zero on every coordinate outside of $V$. An $\epsilon$-extractor for $(n,k)$-zero-fixing sources is a mapping $F:\{0,1\}^n\to\{0,1\}^m$, for some $m$, such that $F(X)$ is $\epsilon$-close in statistical ... more >>>

TR19-057 | 6th April 2019
Olaf Beyersdorff, Joshua Blinkhorn

#### Proof Complexity of Symmetry Learning in QBF

For quantified Boolean formulas (QBF), a resolution system with a symmetry rule was recently introduced by Kauers and Seidl (Inf. Process. Lett. 2018). In this system, many formulas hard for QBF resolution admit short proofs.

Kauers and Seidl apply the symmetry rule on symmetries of the original formula. Here we ... more >>>

TR19-056 | 11th April 2019
Tom Gur, Oded Lachish

#### A Lower Bound for Relaxed Locally Decodable Codes

Revisions: 1

A locally decodable code (LDC) C:{0,1}^k -> {0,1}^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to ... more >>>

TR19-055 | 9th April 2019
Kasper Green Larsen, Tal Malkin, Omri Weinstein, Kevin Yeo

#### Lower Bounds for Oblivious Near-Neighbor Search

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search (ANN) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the ... more >>>

TR19-054 | 9th April 2019
Joshua Brakensiek, Venkatesan Guruswami

#### Bridging between 0/1 and Linear Programming via Random Walks

Under the Strong Exponential Time Hypothesis, an integer linear program with $n$ Boolean-valued variables and $m$ equations cannot be solved in $c^n$ time for any constant $c < 2$. If the domain of the variables is relaxed to $[0,1]$, the associated linear program can of course be solved in polynomial ... more >>>

TR19-053 | 5th April 2019
Andrei Krokhin, Jakub Opršal

#### The complexity of 3-colouring $H$-colourable graphs

We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph $H$, the $H$-colouring problem is to decide whether a given graph has a homomorphism to $H$. By a result of Hell and Nešet?il, this problem is NP-hard for any non-bipartite graph $H$. In ... more >>>

TR19-052 | 9th April 2019
Nicola Galesi, Leszek Kolodziejczyk, Neil Thapen

#### Polynomial calculus space and resolution width

We show that if a $k$-CNF requires width $w$ to refute in resolution, then it requires space $\sqrt w$ to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is ... more >>>

TR19-051 | 9th April 2019
Emanuele Viola

#### Pseudorandom bits and lower bounds for randomized Turing machines

We exhibit a pseudorandom generator with nearly quadratic stretch for randomized Turing machines, which have a one-way random tape and a two-way work tape. This is the first generator for this model. Its stretch is essentially the best possible given current lower bounds. We use the generator to prove a ... more >>>

TR19-050 | 20th March 2019
Titus Dose, Christian Glaßer

#### NP-Completeness, Proof Systems, and Disjoint NP-Pairs

The article investigates the relation between three well-known hypotheses.
1) Hunion: the union of disjoint complete sets for NP is complete for NP
2) Hopps: there exist optimal propositional proof systems
3) Hcpair: there exist complete disjoint NP-pairs

The following results are obtained:
a) The hypotheses are pairwise independent ... more >>>

TR19-049 | 2nd April 2019

#### A Tight Parallel-Repetition Theorem for Random-Terminating Interactive Arguments

Revisions: 1

Parallel repetition is known to reduce the soundness error of some special cases of interactive arguments: three-message protocols and public-coin protocols. However, it does not do so in the general case.

Haitner [FOCS '09, SiCOMP '13] presented a simple method for transforming any interactive argument $\pi$ into a slightly modified ... more >>>

TR19-048 | 2nd April 2019
Per Austrin, Amey Bhangale, Aditya Potukuchi

#### Simplified inpproximability of hypergraph coloring via t-agreeing families

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khachatrian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs ... more >>>

TR19-047 | 2nd April 2019
Mrinal Kumar, Ben Lee Volk

#### Lower Bounds for Matrix Factorization

We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of ... more >>>

TR19-046 | 1st April 2019
Akash Kumar, C. Seshadhri, Andrew Stolman

#### andom walks and forbidden minors II: A $\poly(d\eps^{-1})$-query tester for minor-closed properties of bounded degree graphs

Revisions: 1

Let $G$ be a graph with $n$ vertices and maximum degree $d$. Fix some minor-closed property $\mathcal{P}$ (such as planarity).
We say that $G$ is $\varepsilon$-far from $\mathcal{P}$ if one has to remove $\varepsilon dn$ edges to make it have $\mathcal{P}$.
The problem of property testing $\mathcal{P}$ was introduced in ... more >>>

TR19-045 | 19th February 2019
Jiawei Gao

#### On the Fine-grained Complexity of Least Weight Subsequence in Graphs

Least Weight Subsequence (LWS) is a type of highly sequential optimization problems with form $F(j) = \min_{i < j} [F(i) + c_{i,j}]$. They can be solved in quadratic time using dynamic programming, but it is not known whether these problems can be solved faster than $n^{2-o(1)}$ time. Surprisingly, each such ... more >>>

TR19-044 | 28th March 2019
Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

#### DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>

TR19-043 | 12th March 2019
Toniann Pitassi, Morgan Shirley, Thomas Watson

#### Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity

We study the Boolean Hierarchy in the context of two-party communication complexity, as well as the analogous hierarchy defined with one-sided error randomness instead of nondeterminism. Our results provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove a query-to-communication ... more >>>

TR19-042 | 18th March 2019
Ankit Garg, Nikhil Gupta, Neeraj Kayal, Chandan Saha

#### Determinant equivalence test over finite fields and over $\mathbf{Q}$

The determinant polynomial $Det_n(\mathbf{x})$ of degree $n$ is the determinant of a $n \times n$ matrix of formal variables. A polynomial $f$ is equivalent to $Det_n$ over a field $\mathbf{F}$ if there exists a $A \in GL(n^2,\mathbf{F})$ such that $f = Det_n(A \cdot \mathbf{x})$. Determinant equivalence test over $\mathbf{F}$ is ... more >>>

TR19-041 | 7th March 2019
Srinivasan Arunachalam, Alex Bredariol Grilo, Aarthi Sundaram

#### Quantum hardness of learning shallow classical circuits

In this paper we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of PAC learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results.

1) Hardness of learning ... more >>>

TR19-040 | 19th February 2019
Sanjana Kolisetty, Linh Le, Ilya Volkovich, Mihalis Yannakakis

#### The Complexity of Finding {$S$}-factors in Regular Graphs

A graph $G$ has an \emph{$S$-factor} if there exists a spanning subgraph $F$ of $G$ such that for all $v \in V: \deg_F(v) \in S$.
The simplest example of such factor is a $1$-factor, which corresponds to a perfect matching in a graph. In this paper we study the computational ... more >>>

TR19-039 | 12th March 2019
Eric Allender, Archit Chauhan, Samir Datta, Anish Mukherjee

#### Planarity, Exclusivity, and Unambiguity

We provide new upper bounds on the complexity of the s-t-connectivity problem in planar graphs, thereby providing additional evidence that this problem is not complete for NL. This also yields a new upper bound on the complexity of computing edit distance. Building on these techniques, we provide new upper bounds ... more >>>

TR19-038 | 7th March 2019
Itay Berman, Akshay Degwekar, Ron D. Rothblum, Prashant Nalini Vasudevan

Revisions: 1

The polarization lemma for statistical distance ($\mathrm{SD}$), due to Sahai and Vadhan (JACM, 2003), is an efficient transformation taking as input a pair of circuits $(C_0,C_1)$ and an integer $k$ and outputting a new pair of circuits $(D_0,D_1)$ such that if $\mathrm{SD}(C_0,C_1)\geq\alpha$ then $\mathrm{SD}(D_0,D_1) \geq 1-2^{-k}$ and if $\mathrm{SD}(C_0,C_1) \leq ... more >>> TR19-037 | 5th March 2019 Chi-Ning Chou, Mrinal Kumar, Noam Solomon #### Closure of VP under taking factors: a short and simple proof Revisions: 1 In this note, we give a short, simple and almost completely self contained proof of a classical result of Kaltofen [Kal86, Kal87, Kal89] which shows that if an n variate degree$d$polynomial f can be computed by an arithmetic circuit of size s, then each of its factors can ... more >>> TR19-036 | 5th March 2019 Pavel Hrubes #### On the complexity of computing a random Boolean function over the reals We say that a first-order formula$A(x_1,\dots,x_n)$over$\mathbb{R}$defines a Boolean function$f:\{0,1\}^n\rightarrow\{0,1\}$, if for every$x_1,\dots,x_n\in\{0,1\}$,$A(x_1,\dots,x_n)$is true iff$f(x_1,\dots,x_n)=1$. We show that: (i) every$f$can be defined by a formula of size$O(n)$, (ii) if$A$is required to have at most$k\geq 1$... more >>> TR19-035 | 5th March 2019 Alexey Milovanov #### PIT for depth-4 circuits and Sylvester-Gallai theorem for polynomials This text is a development of a preprint of Ankit Gupta. We present an approach for devising a deterministic polynomial time whitekbox identity testing (PIT) algorithm for depth-$4$circuits with bounded top fanin. This approach is similar to Kayal-Saraf approach for depth-$3$circuits. Kayal and Saraf based their ... more >>> TR19-034 | 5th March 2019 Pavel Hrubes #### On$\epsilon$-sensitive monotone computations We show that strong-enough lower bounds on monotone arithmetic circuits or the non-negative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial$f\in {\mathbb {R}}[x_1,\dots, x_n]$of degree$d$has an arithmetic circuit of size$s$then$(x_1+\dots+x_n+1)^d+\epsilon ... more >>>

TR19-033 | 20th February 2019
Ashish Dwivedi, Rajat Mittal, Nitin Saxena

#### Counting basic-irreducible factors mod $p^k$ in deterministic poly-time and $p$-adic applications

Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The irreducible ... more >>>

TR19-032 | 4th March 2019
Srikanth Srinivasan

#### Strongly Exponential Separation Between Monotone VP and Monotone VNP

We show that there is a sequence of explicit multilinear polynomials $P_n(x_1,\ldots,x_n)\in \mathbb{R}[x_1,\ldots,x_n]$ with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for $P_n$ must have size $\exp(\Omega(n)).$ This builds on (and strengthens) a result of Yehudayoff (2018) who showed a lower bound of $\exp(\tilde{\Omega}(\sqrt{n})).$

more >>>

TR19-031 | 4th March 2019
Lijie Chen

#### Non-deterministic Quasi-Polynomial Time is Average-case Hard for ACC Circuits

Following the seminal work of [Williams, J. ACM 2014], in a recent breakthrough, [Murray and Williams, STOC 2018] proved that NQP (non-deterministic quasi-polynomial time) does not have polynomial-size ACC^0 circuits.

We strengthen the above lower bound to an average case one, by proving that for all constants c, ... more >>>

TR19-030 | 19th February 2019
Claude Crépeau, Nan Yang

#### Non-Locality in Interactive Proofs

In multi-prover interactive proofs (MIPs), the verifier is usually non-adaptive. This stems from an implicit problem which we call “contamination” by the verifier. We make explicit the verifier contamination problem, and identify a solution by constructing a generalization of the MIP model. This new model quantifies non-locality as a new ... more >>>

TR19-029 | 20th February 2019
Yuval Filmus, Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami, David Zuckerman

#### Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions

The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(\frac{n}{\log n})$ players can bias the outcome of *any* Boolean function $\{0,1\}^n \to \{0,1\}$ with respect to the uniform measure. We extend their result to arbitrary product measures on $\{0,1\}^n$, by combining their argument with a completely ... more >>>

TR19-028 | 1st March 2019
Shachar Lovett, Noam Solomon, Jiapeng Zhang

#### From DNF compression to sunflower theorems via regularity

Revisions: 1

The sunflower conjecture is one of the most well-known open problems in combinatorics. It has several applications in theoretical computer science, one of which is DNF compression, due to Gopalan, Meka and Reingold [Computational Complexity 2013]. In this paper, we show that improved bounds for DNF compression imply improved bounds ... more >>>

TR19-027 | 1st March 2019
Mark Bun, Nikhil Mande, Justin Thaler

#### Sign-Rank Can Increase Under Intersection

The communication class $UPP^{cc}$ is a communication analog of the Turing Machine complexity class $PP$. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.

For a communication problem ... more >>>

TR19-026 | 28th February 2019
Pavel Hrubes, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, Amir Yehudayoff

Revisions: 1

There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets $S_1,\ldots,S_k \subset [n]$ is balancing if for every subset $X \subset \{1,2,\ldots,n\}$ of size $n/2$, there is an $i \in [k]$ so that $|S_i \cap X| = ... more >>> TR19-025 | 28th February 2019 Shuichi Hirahara, Osamu Watanabe #### On Nonadaptive Reductions to the Set of Random Strings and Its Dense Subsets We investigate the computational power of an arbitrary distinguisher for (not necessarily computable) hitting set generators as well as the set of Kolmogorov-random strings. This work contributes to (at least) two lines of research. One line of research is the study of the limits of black-box reductions to some distributional ... more >>> TR19-024 | 20th February 2019 Russell Impagliazzo, Sasank Mouli, Toniann Pitassi #### The Surprising Power of Constant Depth Algebraic Proofs A major open problem in proof complexity is to prove super-polynomial lower bounds for AC^0[p]-Frege proofs. This system is the analog of AC^0[p], the class of bounded depth circuits with prime modular counting gates. Despite strong lower bounds for this class dating back thirty years (Razborov, '86 and Smolensky, '87), ... more >>> TR19-023 | 25th February 2019 Orr Paradise #### Smooth and Strong PCPs Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: - ... more >>> TR19-022 | 23rd February 2019 Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis #### Circuit Lower Bounds for MCSP from Local Pseudorandom Generators The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function$f$can be computed by a Boolean circuit of size at most$\theta$, for a given parameter$\theta$. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a ... more >>> TR19-021 | 19th February 2019 Rahul Ilango ####$AC^0[p]$Lower Bounds and NP-Hardness for Variants of MCSP The Minimum Circuit Size Problem (MCSP) asks whether a (given) Boolean function has a circuit of at most a (given) size. Despite over a half-century of study, we know relatively little about the computational complexity of MCSP. We do know that questions about the complexity of MCSP have significant ramifications ... more >>> TR19-020 | 4th February 2019 Ludmila Glinskih, Dmitry Itsykson #### On Tseitin formulas, read-once branching programs and treewidth Revisions: 1 We show that any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on an$n\times n$grid graph has size at least$2^{\Omega(n)}$. Then using the Excluded Grid Theorem by Robertson and Seymour we show that for arbitrary graph$G(V,E)$any nondeterministic read-once branching program that computes ... more >>> TR19-019 | 19th February 2019 Mrinal Kumar, Rafael Mendes de Oliveira, Ramprasad Saptharishi #### Towards Optimal Depth Reductions for Syntactically Multilinear Circuits We show that any$n$-variate polynomial computable by a syntactically multilinear circuit of size$\mathop{poly}(n)$can be computed by a depth-$4$syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at most$\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree$d = \omega(n/\log n)$, this improves upon the upper bound of$\exp\left({O(\sqrt{d}\log n)}\right)$obtained by Tavenas (MFCS ... more >>> TR19-018 | 18th February 2019 Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Avishay Tal #### AC0[p] Lower Bounds against MCSP via the Coin Problem Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an$n$-variate boolean function has circuit complexity less than a given parameter$s$. We prove that MCSP is hard for constant-depth circuits with mod$p$gates, for any prime$p\geq 2$(the circuit class$AC^0[p])$. Namely, ... more >>> TR19-017 | 6th February 2019 Chin Ho Lee #### Fourier bounds and pseudorandom generators for product tests We study the Fourier spectrum of functions$f\colon \{0,1\}^{mk} \to \{-1,0,1\}$which can be written as a product of$k$Boolean functions$f_i$on disjoint$m$-bit inputs. We prove that for every positive integer$d$, $\sum_{S \subseteq [mk]: |S|=d} |\hat{f_S}| = O(m)^d .$ Our upper bound ... more >>> TR19-016 | 5th February 2019 Alexander A. Sherstov #### The hardest halfspace We study the approximation of halfspaces$h:\{0,1\}^n\to\{0,1\}$in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the "hardest" halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all ... more >>> TR19-015 | 7th February 2019 William Kretschmer #### QMA Lower Bounds for Approximate Counting We prove a query complexity lower bound for$QMA$protocols that solve approximate counting: estimating the size of a set given a membership oracle. This gives rise to an oracle$A$such that$SBP^A \not\subset QMA^A$, resolving an open problem of Aaronson [2]. Our proof uses the polynomial method to ... more >>> TR19-014 | 22nd January 2019 Himanshu Tyagi, Shun Watanabe #### A New Proof of Nonsignalling Multiprover Parallel Repetition Theorem We present an information theoretic proof of the nonsignalling multiprover parallel repetition theorem, a recent extension of its two-prover variant that underlies many hardness of approximation results. The original proofs used de Finetti type decomposition for strategies. We present a new proof that is based on a technique we introduced ... more >>> TR19-013 | 31st January 2019 Joshua Brakensiek, Sivakanth Gopi, Venkatesan Guruswami #### CSPs with Global Modular Constraints: Algorithms and Hardness via Polynomial Representations We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo$M$, for various choices of the modulus$M$. Due to the known classification of tractable Boolean CSPs, this mainly reduces to the study of three cases: 2SAT, HornSAT, ... more >>> TR19-012 | 27th January 2019 Oded Goldreich #### Multi-pseudodeterministic algorithms In this work, dedicated to Shafi Goldwasser, we consider a relaxation of the notion of pseudodeterministic algorithms, which was put forward by Gat and Goldwasser ({\em ECCC}, TR11--136, 2011). Pseudodeterministic algorithms are randomized algorithms that solve search problems by almost always providing the same canonical solution (per each input). ... more >>> TR19-011 | 27th January 2019 Benny Applebaum, Eliran Kachlon #### Sampling Graphs without Forbidden Subgraphs and Almost-Explicit Unbalanced Expanders We initiate the study of the following hypergraph sampling problem: Sample a$d$-uniform hypergraph over$n$vertices and$m$hyperedges from some pseudorandom distribution$\mathcal{G}$conditioned on not having some small predefined$t$-size hypergraph$H$as a subgraph. The algorithm should run in$\mathrm{poly}(n)$-time even when the size of the ... more >>> TR19-010 | 21st January 2019 Dorit Aharonov, Alex Bredariol Grilo #### Stoquastic PCP vs. Randomness The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this work, we connect this problem to a variant of another major problem: the quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the ... more >>> TR19-009 | 16th January 2019 Jiawei Gao, Russell Impagliazzo #### The Fine-Grained Complexity of Strengthenings of First-Order Logic Revisions: 1 The class of model checking for first-order formulas on sparse graphs has a complete problem with respect to fine-grained reductions, Orthogonal Vectors (OV) [GIKW17]. This paper studies extensions of this class or more lenient parameterizations. We consider classes obtained by allowing function symbols; first-order on ordered structures; adding various notions ... more >>> TR19-008 | 20th January 2019 Ashish Dwivedi, Rajat Mittal, Nitin Saxena #### Efficiently factoring polynomials modulo$p^4$Polynomial factoring has famous practical algorithms over fields-- finite, rational \&$p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example,$x^2+p \bmod p^2$is irreducible, but$x^2+px \bmod p^2$has exponentially many factors! We present the first randomized poly($\deg ... more >>>

TR19-007 | 17th January 2019
Arkadev Chattopadhyay, Meena Mahajan, Nikhil Mande, Nitin Saurabh

#### Lower Bounds for Linear Decision Lists

We demonstrate a lower bound technique for linear decision lists, which are decision lists where the queries are arbitrary linear threshold functions.
We use this technique to prove an explicit lower bound by showing that any linear decision list computing the function $MAJ \circ XOR$ requires size $2^{0.18 n}$. This ... more >>>

TR19-006 | 17th January 2019
Anna Gal, Ridwan Syed

#### Upper Bounds on Communication in terms of Approximate Rank

Revisions: 1

We show that any Boolean function with approximate rank $r$ can be computed by bounded error quantum protocols without prior entanglement of complexity $O( \sqrt{r} \log r)$. In addition, we show that any Boolean function with approximate rank $r$ and discrepancy $\delta$ can be computed by deterministic protocols of complexity ... more >>>

TR19-005 | 16th January 2019
Omar Alrabiah, Venkatesan Guruswami

#### An Exponential Lower Bound on the Sub-Packetization of MSR Codes

An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a ... more >>>

TR19-004 | 11th January 2019
Amey Bhangale, Subhash Khot

#### UG-hardness to NP-hardness by Losing Half

The $2$-to-$2$ Games Theorem of [KMS-1, DKKMS-1, DKKMS-2, KMS-2] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least $(\frac{1}{2}-\varepsilon)$ fraction of the constraints $vs.$ no assignment satisfying more than $\varepsilon$ fraction of the constraints, for every constant $\varepsilon>0$. We show that the reduction ... more >>>

TR19-003 | 2nd January 2019
Alexander A. Sherstov, Pei Wu

#### Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0

The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a real matrix $M$ with $\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}$. Determining the maximum ... more >>>

TR19-002 | 31st December 2018
Alexander Kulikov, Ivan Mikhailin, Andrey Mokhov, Vladimir Podolskii

#### Complexity of Linear Operators

Let $A \in \{0,1\}^{n \times n}$ be a matrix with $z$ zeroes and $u$ ones and $x$ be an $n$-dimensional vector of formal variables over a semigroup $(S, \circ)$. How many semigroup operations are required to compute the linear operator $Ax$?

As we observe in this paper, this problem contains ... more >>>

TR19-001 | 5th January 2019
Dmitry Itsykson, Alexander Knop, Andrei Romashchenko, Dmitry Sokolov

#### On OBDD-based algorithms and proof systems that dynamically change order of variables

In 2004 Atserias, Kolaitis and Vardi proposed OBDD-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of identically false OBDD from OBDDs representing clauses of the initial formula. All OBDDs in such proofs have the same order of variables. We initiate the study of OBDD based ... more >>>

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