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REPORTS > KEYWORD > LOWER BOUNDS:
Reports tagged with lower bounds:
TR12-041 | 17th April 2012
Stasys Jukna

#### Limitations of Incremental Dynamic Programs

Revisions: 1

We consider so-called incremental'' dynamic programming (DP) algorithms, and are interested in the number of subproblems produced by them. The standard DP algorithm for the n-dimensional Knapsack problem is incremental, and produces nK subproblems, where K is the capacity of the knapsack. We show that any incremental algorithm for this ... more >>>

TR16-168 | 2nd November 2016
Eric Blais, Clement Canonne, Tom Gur

#### Alice and Bob Show Distribution Testing Lower Bounds (They don't talk to each other anymore.)

Revisions: 1

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [BBM12], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method ... more >>>

TR16-187 | 21st November 2016
morris yau

#### Almost Cubic Bound for Depth Three Circuits in VP

Revisions: 3

In "An Almost Cubic Lower Bound for $\sum\prod\sum$ circuits in VP", [BLS16] present an infinite family of polynomials, $\{P_n\}_{n \in \mathbb{Z}^+}$, with $P_n$
on $N = \Theta(n polylog(n))$
variables with degree $N$ being in VP such that every
$\sum\prod\sum$ circuit computing $P_n$ is of size $\Omega\big(\frac{N^3}{2^{\sqrt{\log N}}}\big)$.
We ... more >>>

TR16-188 | 21st November 2016
Toniann Pitassi, Robert Robere

#### Strongly Exponential Lower Bounds for Monotone Computation

For a universal constant $\alpha > 0$, we prove size lower bounds of $2^{\alpha N}$ for computing an explicit monotone function in NP in the following models of computation: monotone formulas, monotone switching networks, monotone span programs, and monotone comparator circuits, where $N$ is the number of variables of the ... more >>>

TR17-020 | 12th February 2017
Ran Raz

#### A Time-Space Lower Bound for a Large Class of Learning Problems

We prove a general time-space lower bound that applies for a large class of learning problems and shows that for every problem in that class, any learning algorithm requires either a memory of quadratic size or an exponential number of samples.

Our result is stated in terms of the norm ... more >>>

TR17-022 | 13th February 2017
Benjamin Rossman, Srikanth Srinivasan

#### Separation of AC$^0[\oplus]$ Formulas and Circuits

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent ... more >>>

TR17-028 | 17th February 2017
Mrinal Kumar

#### A quadratic lower bound for homogeneous algebraic branching programs

Revisions: 1

An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex $s$, and end vertex $t$ and each edge having a weight which is an affine form in $\F[x_1, x_2, \ldots, x_n]$. An ABP computes a polynomial in a natural way, as the sum of weights of ... more >>>

TR17-032 | 17th February 2017

#### Formulas with Large Weight: a New Technique for Genuine QBF Lower Bounds

We devise a new technique to prove lower bounds for the proof size in resolution-type calculi for quantified Boolean formulas (QBF). The new technique applies to the strong expansion system IR-calc and thereby also to the most studied QBF system Q-Resolution.

Our technique exploits a clear semantic paradigm, showing the ... more >>>

TR17-044 | 21st February 2017
Olaf Beyersdorff, Luke Hinde, Ján Pich

#### Reasons for Hardness in QBF Proof Systems

Revisions: 1

We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction.

The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS'16). Here we ... more >>>

TR17-121 | 31st July 2017
Sumegha Garg, Ran Raz, Avishay Tal

#### Extractor-Based Time-Space Lower Bounds for Learning

Revisions: 1

A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem: An unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen ... more >>>

TR17-150 | 26th September 2017
Andris Ambainis, Martins Kokainis, Krisjanis Prusis, Jevgenijs Vihrovs

#### All Classical Adversary Methods are Equivalent for Total Functions

Revisions: 2

We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity $\text{fbs}(f)$. That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. For partial functions, we show ... more >>>

TR17-162 | 26th October 2017
Klim Efremenko, Ankit Garg, Rafael Mendes de Oliveira, Avi Wigderson

#### Barriers for Rank Methods in Arithmetic Complexity

Arithmetic complexity, the study of the cost of computing polynomials via additions and multiplications, is considered (for many good reasons) simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than ... more >>>

TR17-165 | 3rd November 2017
Toniann Pitassi, Robert Robere

#### Lifting Nullstellensatz to Monotone Span Programs over Any Field

We characterize the size of monotone span programs computing certain "structured" boolean functions by the Nullstellensatz degree of a related unsatisfiable Boolean formula.

This yields the first exponential lower bounds for monotone span programs over arbitrary fields, the first exponential separations between monotone span programs over fields of different ... more >>>

TR17-193 | 31st December 2017
Oded Goldreich, Avishay Tal

#### On Constant-Depth Canonical Boolean Circuits for Computing Multilinear Functions

We consider new complexity measures for the model of multilinear circuits with general multilinear gates introduced by Goldreich and Wigderson (ECCC, 2013).
These complexity measures are related to the size of canonical constant-depth Boolean circuits, which extend the definition of canonical depth-three Boolean circuits.
We obtain matching lower and upper ... more >>>

TR18-020 | 30th January 2018
Meena Mahajan, Prajakta Nimbhorkar, Anuj Tawari

#### Computing the maximum using $(\min, +)$ formulas

We study computation by formulas over $(min, +)$. We consider the computation of $\max\{x_1,\ldots,x_n\}$
over $\mathbb{N}$ as a difference of $(\min, +)$ formulas, and show that size $n + n \log n$ is sufficient and necessary. Our proof also shows that any $(\min, +)$ formula computing the minimum of all ... more >>>

TR18-114 | 6th June 2018
Paul Beame, Shayan Oveis Gharan, Xin Yang

#### Time-Space Tradeoffs for Learning Finite Functions from Random Evaluations, with Applications to Polynomials

We develop an extension of recent analytic methods for obtaining time-space tradeoff lower bounds for problems of learning from uniformly random labelled examples. With our methods we can obtain bounds for learning concept classes of finite functions from random evaluations even when the sample space of random inputs can be ... more >>>

TR18-146 | 18th August 2018
Meena Mahajan, Prajakta Nimbhorkar, Anuj Tawari

#### Shortest path length with bounded-alternation $(\min, +)$ formulas

We study bounded depth $(\min, +)$ formulas computing the shortest path polynomial. For depth $2d$ with $d \geq 2$, we obtain lower bounds parametrized by certain fan-in restrictions on all $+$ gates except those at the bottom level. For depth $4$, in two regimes of the parameter, the bounds are ... 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 >>>

TR19-071 | 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 >>>

TR19-073 | 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 >>>

TR19-100 | 31st July 2019
Hervé Fournier, Guillaume Malod, Maud Szusterman, Sébastien Tavenas

#### Nonnegative rank measures and monotone algebraic branching programs

Inspired by Nisan's characterization of noncommutative complexity (Nisan 1991), we study different notions of nonnegative rank, associated complexity measures and their link with monotone computations. In particular we answer negatively an open question of Nisan asking whether nonnegative rank characterizes monotone noncommutative complexity for algebraic branching programs. We also prove ... more >>>

TR19-128 | 24th September 2019
Anna Gal, Robert Robere

#### Lower Bounds for (Non-monotone) Comparator Circuits

Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make "copies" of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and ... more >>>

TR19-173 | 28th November 2019
Divesh Aggarwal, Siyao Guo, Maciej Obremski, Joao Ribeiro, Noah Stephens-Davidowitz

#### Extractor Lower Bounds, Revisited

Revisions: 1

We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a change in quantifiers'' over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input ... more >>>

TR19-174 | 2nd December 2019
Susanna de Rezende, Jakob Nordström, Kilian Risse, Dmitry Sokolov

#### Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense ... 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 >>>

TR20-012 | 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 >>>

TR20-034 | 12th March 2020
Erfan Khaniki

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

Revisions: 3

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 >>>

TR20-039 | 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 >>>

TR20-053 | 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 >>>

TR20-060 | 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 >>>

TR20-062 | 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 >>>

TR20-073 | 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 >>>

TR20-123 | 17th August 2020

#### An Optimal Tester for k-Linear

A Boolean function $f:\{0,1\}^n\to \{0,1\}$ is $k$-linear if it returns the sum (over the binary field $F_2$) of $k$ coordinates of the input. In this paper, we study property testing of the classes $k$-Linear, the class of all $k$-linear functions, and $k$-Linear$^*$, the class $\cup_{j=0}^kj$-Linear.
We give a non-adaptive distribution-free ... 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 >>>

TR21-076 | 4th June 2021
Dmitry Sokolov

#### Pseudorandom Generators, Resolution and Heavy Width

Revisions: 1

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 >>>

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

#### The Space Complexity of Sampling

Revisions: 1

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 >>>

TR21-159 | 15th November 2021
Lijie Chen, Ce Jin, Rahul Santhanam, Ryan Williams

#### Constructive Separations and Their Consequences

For a complexity class $C$ and language $L$, a constructive separation of $L \notin C$ gives an efficient algorithm (also called a refuter) to find counterexamples (bad inputs) for every $C$-algorithm attempting to decide $L$. We study the questions: Which lower bounds can be made constructive? What are the consequences ... more >>>

TR22-040 | 10th March 2022
Benjamin Böhm, Tomáš Peitl, Olaf Beyersdorff

#### Should decisions in QCDCL follow prefix order?

Quantified conflict-driven clause learning (QCDCL) is one of the main solving approaches for quantified Boolean formulas (QBF). One of the differences between QCDCL and propositional CDCL is that QCDCL typically follows the prefix order of the QBF for making decisions.
We investigate an alternative model for QCDCL solving where decisions ... more >>>

TR22-071 | 13th May 2022
We establish an $\epsilon$-sensitive hierarchy separation for monotone arithmetic computations. The notion of $\epsilon$-sensitive monotone lower bounds was recently introduced by Hrubes [Computational Complexity'20]. We show the following:
(1) There exists a monotone polynomial over $n$ variables in VNP that cannot be computed by $2^{o(n)}$ size monotone ... more >>>