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TR19-132 | 26th September 2019
Klim Efremenko, Gillat Kol, Raghuvansh Saxena

We consider the celebrated radio network model for abstracting communication in wireless networks. In this model, in any round, each node in the network may broadcast a message to all its neighbors. However, a node is able to hear a message broadcast by a neighbor only if no collision occurred, ... more >>>

TR19-094 | 16th July 2019
Venkatesan Guruswami, Sai Sandeep

#### Rainbow coloring hardness via low sensitivity polymorphisms

A $k$-uniform hypergraph is said to be $r$-rainbow colorable if there is an $r$-coloring of its vertices such that every hyperedge intersects all $r$ color classes. Given as input such a hypergraph, finding a $r$-rainbow coloring of it is NP-hard for all $k \ge 3$ and $r \ge 2$. ... more >>>

TR06-043 | 22nd March 2006
Eran Ofek, Uriel Feige

#### Random 3CNF formulas elude the Lovasz theta function

Let $\phi$ be a 3CNF formula with n variables and m clauses. A
simple nonconstructive argument shows that when m is
sufficiently large compared to n, most 3CNF formulas are not
satisfiable. It is an open question whether there is an efficient
refutation algorithm that for most such formulas proves ... more >>>

TR12-033 | 5th April 2012
Ankit Gupta, Neeraj Kayal, Youming Qiao

#### Random Arithmetic Formulas can be Reconstructed Efficiently

Informally stated, we present here a randomized algorithm that given blackbox access to the polynomial $f$ computed by an unknown/hidden arithmetic formula $\phi$ reconstructs, on the average, an equivalent or smaller formula $\hat{\phi}$ in time polynomial in the size of its output $\hat{\phi}$.

Specifically, we consider arithmetic formulas wherein the ... more >>>

TR17-045 | 7th March 2017
Noah Fleming, Denis Pankratov, Toniann Pitassi, Robert Robere

#### Random CNFs are Hard for Cutting Planes

Revisions: 2

The random k-SAT model is the most important and well-studied distribution over
k-SAT instances. It is closely connected to statistical physics; it is used as a testbench for
satisfiablity algorithms, and lastly average-case hardness over this distribution has also
been linked to hardness of approximation via Feige’s hypothesis. In this ... more >>>

TR09-137 | 14th December 2009
Massimo Lauria

#### Random CNFs require spacious Polynomial Calculus refutations

We study the space required by Polynomial Calculus refutations of random $k$-CNFs. We are interested in how many monomials one needs to keep in memory to carry on a refutation. More precisely we show that for $k \geq 4$ a refutation of a random $k$-CNF of $\Delta n$ clauses and ... more >>>

TR17-042 | 6th March 2017
Pavel Hrubes, Pavel Pudlak

#### Random formulas, monotone circuits, and interpolation

We prove new lower bounds on the sizes of proofs in the Cutting Plane proof system, using a concept that we call "unsatisfiability certificate". This approach is, essentially, equivalent to the well-known feasible interpolation method, but is applicable to CNF formulas that do not seem suitable for interpolation. Specifically, we ... more >>>

TR09-033 | 16th April 2009
Phokion G. Kolaitis, Swastik Kopparty

#### Random Graphs and the Parity Quantifier

The classical zero-one law for first-order logic on random graphs says that for every first-order property $\varphi$ in the theory of graphs and every $p \in (0,1)$, the probability that the random graph $G(n, p)$ satisfies $\varphi$ approaches either $0$ or $1$ as $n$ approaches infinity. It is well known ... more >>>

TR08-080 | 3rd July 2008
Ido Ben-Eliezer, Rani Hod, Shachar Lovett

#### Random low degree polynomials are hard to approximate

Revisions: 1

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over $\F$.
We prove that, with very high probability, a random degree $d$ polynomial has only an exponentially small correlation with all polynomials of degree $d-1$, for all degrees $d$ up to ... more >>>

TR02-006 | 8th November 2001
Philippe Moser

#### Random nondeterministic real functions and Arthur Merlin games

Revisions: 1

We construct a nondeterministic analogue to \textbf{APP}, denoted
\textbf{NAPP}; which is the set of all real valued functions
$f: \{ 0,1 \}^{*} \rightarrow [0,1]$, that are approximable within 1/$k$,
by a probabilistic nondeterministic transducer, in time poly($n,1^{k}$).
We show that the subset of all Boolean ... more >>>

TR16-175 | 8th November 2016
Pavel Pudlak, Neil Thapen

#### Random resolution refutations

Revisions: 2

We study the \emph{random resolution} refutation system defined in~[Buss et al. 2014]. This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the ... more >>>

TR19-165 | 18th November 2019
Clement Canonne, Xi Chen, Gautam Kamath, Amit Levi, Erik Waingarten

#### Random Restrictions of High-Dimensional Distributions and Uniformity Testing with Subcube Conditioning

We give a nearly-optimal algorithm for testing uniformity of distributions supported on $\{-1,1\}^n$, which makes $\tilde O (\sqrt{n}/\varepsilon^2)$ queries to a subcube conditional sampling oracle (Bhattacharyya and Chakraborty (2018)). The key technical component is a natural notion of random restriction for distributions on $\{-1,1\}^n$, and a quantitative analysis of how ... more >>>

TR01-100 | 14th December 2001
Noga Alon, Wenceslas Fernandez de la Vega, Ravi Kannan, Marek Karpinski

#### Random Sampling and Approximation of MAX-CSP Problems

We present a new efficient sampling method for approximating
r-dimensional Maximum Constraint Satisfaction Problems, MAX-rCSP, on
n variables up to an additive error \epsilon n^r.We prove a new
general paradigm in that it suffices, for a given set of constraints,
to pick a small uniformly random ... more >>>

TR06-026 | 27th February 2006

#### Random Selection with an Adversarial Majority

We consider the problem of random selection, where $p$ players follow a protocol to jointly select a random element of a universe of size $n$. However, some of the players may be adversarial and collude to force the output to lie in a small subset of the universe. We describe ... more >>>

TR98-049 | 10th July 1998
Dimitris Fotakis, Paul Spirakis

#### Random Walks, Conditional Hitting Sets and Partial Derandomization

In this work we use random walks on expanders in order to
relax the properties of hitting sets required for partially
derandomizing one-side error algorithms. Building on a well-known
probability amplification technique [AKS87,CW89,IZ89], we use
random walks on expander graphs of subexponential (in the
random bit complexity) size so as ... more >>>

TR05-065 | 26th June 2005
Alexander Barvinok, Alex Samorodnitsky

#### Random Weighting, Asymptotic Counting, and Inverse Isoperimetry

For a family X of k-subsets of the set 1...n, let |X| be the cardinality of X and let Gamma(X, mu) be the expected maximum weight of a subset from X when the weights of 1...n are chosen independently at random from a symmetric probability distribution mu on R. We ... more >>>

TR10-056 | 1st April 2010
Kord Eickmeyer, Martin Grohe

#### Randomisation and Derandomisation in Descriptive Complexity Theory

Revisions: 1

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic $\mathcal{L}$ we introduce a new logic $\mathsf{BP}\mathcal{L}$, bounded error probabilistic $\mathcal{L}$, which is defined from $\mathcal{L}$ in a similar way as the
complexity class $\mathsf{BPP}$, bounded error probabilistic polynomial time, is defined ... more >>>

TR97-021 | 16th May 1997
Farid Ablayev

#### Randomization and nondeterminsm are incomparable for ordered read-once branching programs

In the manuscript F. Ablayev and M. Karpinski, On the power of
randomized branching programs (generalization of ICALP'96 paper
results for the case of pure boolean function, available at
http://www.ksu.ru/~ablayev) we exhibited a simple boolean functions
$f_n$ in $n$ variables such that:

1) $f_{n}$ can be computed ... more >>>

TR96-058 | 25th November 1996
Dima Grigoriev, Marek Karpinski

#### Randomized $\mathbf{\Omega (n^2)}$ Lower Bound for Knapsack

We prove $\Omega (n^2)$ complexity \emph{lower bound} for the
general model of \emph{randomized computation trees} solving
the \emph{Knapsack Problem}, and more generally \emph{Restricted
Integer Programming}. This is the \emph{first nontrivial} lower
bound proven for this model of computation. The method of the ... more >>>

TR00-007 | 14th December 1999
Pavlos S. Efraimidis, Paul Spirakis

#### Randomized Approximation Schemes for Scheduling Unrelated Parallel Machines

The problem of Scheduling $n$ Independent Jobs
on $m$ Unrelated Parallel Machines, when $m$
is fixed, is considered. The standard problem
of minimizing the makespan of the schedule
(SUM) and the bicriteria problem of scheduling
with bounded makespan and cost (SUMC), are
addressed, and randomized fully linear time
more >>>

TR13-178 | 14th December 2013
Nikolay Vereshchagin

#### Randomized communication complexity of appropximating Kolmogorov complexity

Revisions: 2

The paper [Harry Buhrman, Michal Koucky, Nikolay Vereshchagin. Randomized Individual Communication Complexity. IEEE Conference on Computational Complexity 2008: 321-331] considered communication complexity of the following problem. Alice has a binary string $x$ and Bob a binary string $y$, both of length $n$, and they want to compute or approximate
more >>>

TR15-169 | 23rd October 2015
Mika Göös, T.S. Jayram, Toniann Pitassi, Thomas Watson

#### Randomized Communication vs. Partition Number

Revisions: 1

We show that \emph{randomized} communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal \emph{randomized} lower bounds for the Clique vs.\ Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (G\"o\"os, Pitassi, and Watson, {\small FOCS~2015}).

more >>>

TR99-020 | 9th June 1999
Marek Karpinski

#### Randomized Complexity of Linear Arrangements and Polyhedra

We survey some of the recent results on the complexity of recognizing
n-dimensional linear arrangements and convex polyhedra by randomized
algebraic decision trees. We give also a number of concrete applications
of these results. In particular, we derive first nontrivial, in fact

TR16-089 | 2nd June 2016
Vikraman Arvind, Partha Mukhopadhyay, Raja S

#### Randomized Polynomial Time Identity Testing for Noncommutative Circuits

Revisions: 2

In this paper we show that polynomial identity testing for
noncommutative circuits of size $s$, computing a polynomial in
$\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$, can be done by a randomized algorithm
with running time polynomial in $s$ and $n$. This answers a question
that has been open for over ten years.

The ... more >>>

TR16-087 | 30th May 2016
Shalev Ben-David, Robin Kothari

#### Randomized query complexity of sabotaged and composed functions

We study the composition question for bounded-error randomized query complexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Theta(log R(g)), in between f and g allows us to prove R(f ... more >>>

TR04-059 | 21st June 2004
Beatrice List, Markus Maucher, Uwe Schöning, Rainer Schuler

#### Randomized Quicksort and the Entropy of the Random Number Generator

The worst-case complexity of an implementation of Quicksort depends
on the random number generator that is used to select the pivot
elements. In this paper we estimate the expected number of
comparisons of Quicksort as a function in the entropy of the random
source. We give upper and lower bounds ... more >>>

TR04-009 | 22nd January 2004
Martin Dyer, Alan Frieze, Thomas P. Hayes, Eric Vigoda

#### Randomly coloring constant degree graphs

We study a simple Markov chain, known as the Glauber dynamics, for generating a random <i>k</i>-coloring of a <i>n</i>-vertex graph with maximum degree &Delta;. We prove that the dynamics converges to a random coloring after <i>O</i>(<i>n</i> log <i>n</i>) steps assuming <i>k</i> &ge; <i>k</i><sub>0</sub> for some absolute constant <i>k</i><sub>0</sub>, and either: ... 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 >>>

TR98-018 | 27th March 1998
Martin Sauerhoff

#### Randomness and Nondeterminism are Incomparable for Read-Once Branching Programs

We extend the tools for proving lower bounds for randomized branching
programs by presenting a new technique for the read-once case which is
applicable to a large class of functions. This technique fills the gap
between simple methods only applicable for OBDDs and the well-known
"rectangle technique" of Borodin, Razborov ... more >>>

TR10-175 | 14th November 2010
Emanuele Viola

#### Randomness buys depth for approximate counting

Revisions: 1

We show that the promise problem of distinguishing $n$-bit strings of hamming weight $\ge 1/2 + \Omega(1/\log^{d-1} n)$ from strings of weight $\le 1/2 - \Omega(1/\log^{d-1} n)$ can be solved by explicit, randomized (unbounded-fan-in) poly(n)-size depth-$d$ circuits with error $\le 1/3$, but cannot be solved by deterministic poly(n)-size depth-$(d+1)$ circuits, ... more >>>

TR16-018 | 3rd February 2016
Kuan Cheng, Xin Li

#### Randomness Extraction in $AC^0$ and with Small Locality

Revisions: 7

We study two variants of seeded randomness extractors. The first one, as studied by Goldreich et al. \cite{goldreich2015randomness}, is seeded extractors that can be computed by $AC^0$ circuits. The second one, as introduced by Bogdanov and Guo \cite{bogdanov2013sparse}, is (strong) extractor families that consist of sparse transformations, i.e., functions that ... more >>>

TR13-120 | 4th September 2013
Zeyu Guo

#### Randomness-efficient Curve Samplers

Curve samplers are sampling algorithms that proceed by viewing the domain as a vector space over a finite field, and randomly picking a low-degree curve in it as the sample. Curve samplers exhibit a nice property besides the sampling property: the restriction of low-degree polynomials over the domain to the ... more >>>

TR06-058 | 25th April 2006
Alexander Healy

#### Randomness-Efficient Sampling within NC^1

Revisions: 1

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in AC^0[mod 2]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC^1. For example, we obtain the following results:

... more >>>

TR12-003 | 13th December 2011
Pratik Worah

#### Rank Bounds for a Hierarchy of Lov\'{a}sz and Schrijver

Lov\'{a}sz and Schrijver introduced several lift and project methods for $0$-$1$ integer programs, now collectively known as Lov\'{a}sz-Schrijver ($LS$) hierarchies. Several lower bounds have since been proven for the rank of various linear programming relaxations in the $LS$ and $LS_+$ hierarchies. In this paper we investigate rank bounds in the ... more >>>

TR10-149 | 22nd September 2010
Boaz Barak, Zeev Dvir, Avi Wigderson, Amir Yehudayoff

#### Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

Revisions: 1

A $(q,k,t)$-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most $q$ non-zeros, each column has at least $k$ non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank ... more >>>

TR95-018 | 27th March 1995
Jay Belanger, Jie Wang

#### Rankable Distributions Do Not Provide Harder Instances Than Uniform Distributions

We show that polynomially rankable distributions
do not provide harder instances than uniform distributions
for NP problems. In particular, we show that if Levin's
randomized tiling problem is solvable in polynomial time on
average, then every NP problem under any p-rankable
... more >>>

TR17-010 | 18th January 2017
Xiaodi Wu, Penghui Yao, Henry Yuen

Revisions: 1

\log_2 \log_2 n$have zero correlation with parity. Such a result is false for modular and threshold polynomials. Its proof ... more >>> TR10-158 | 31st October 2010 Shiva Kintali #### Realizable Paths and the NL vs L Problem Revisions: 2 A celebrated theorem of Savitch states that NSPACE(S) is contained DSPACE(S^2). In particular, Savitch gave a deterministic algorithm to solve ST-CONNECTIVITY (an NL-complete problem) using O(log^2{n}) space, implying NL is in DSPACE(log^2{n}). While Savitch’s theorem itself has not been improved in the last four decades, studying the space complexity of ... 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 >>> TR14-041 | 31st March 2014 Shachar Lovett #### Recent advances on the log-rank conjecture in communication complexity The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix, up to polynomial factors. Despite much research, we still know very little about this ... more >>> TR18-151 | 29th August 2018 Ankit Garg, Rafael Oliveira #### Recent progress on scaling algorithms and applications Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer science to telephone forecasting, economics, statistics, optimization, among many other fields. Recently, a generalization of matrix more >>> TR03-062 | 10th July 2003 Andrei Krokhin, Peter Jonsson #### Recognizing Frozen Variables in Constraint Satisfaction Problems In constraint satisfaction problems over finite domains, some variables can be frozen, that is, they take the same value in all possible solutions. We study the complexity of the problem of recognizing frozen variables with restricted sets of constraint relations allowed in the instances. We show that the complexity of ... more >>> TR05-008 | 11th December 2004 Neeraj Kayal #### Recognizing permutation functions in polynomial time. Let$\mathbb{F}_q$be a finite field and$f(x) \in \mathbb{F}_q(x)$be a rational function over$\mathbb{F}_q$. The decision problem {\bf PermFunction} consists of deciding whether$f(x)$induces a permutation on the elements of$\mathbb{F}_q$. That is, we want to decide whether the corresponding map$f : \mathbb{F}_q ... more >>>

TR09-119 | 17th November 2009
Frederic Magniez, Claire Mathieu, Ashwin Nayak

#### Recognizing well-parenthesized expressions in the streaming model

Motivated by a concrete problem and with the goal of understanding the sense in which the complexity of streaming algorithms is related to the complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with $s$ different types of parenthesis.

We present a one-pass randomized streaming ... more >>>

TR15-150 | 13th September 2015
Gaurav Sinha

#### Reconstruction of $\Sigma\Pi\Sigma(2)$ Circuits over Reals

Revisions: 3

Reconstruction of arithmertic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In this paper we present a polynomial time randomized algorithm for reconstructing $\Sigma\Pi\Sigma(2)$ circuits over $\R$, i.e. depth$-3$ circuits with fan-in $2$ at the top addition ... more >>>

TR19-104 | 6th August 2019
Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich

#### Reconstruction of Depth-$4$ Multilinear Circuits

We present a deterministic algorithm for reconstructing multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuits, i.e. multilinear depth-$4$ circuits with fan-in $k$ at the top $+$ gate. For any fixed $k$, given black-box access to a polynomial $f \in \mathbb{F}[x_{1},x_{2},\ldots ,x_{n}]$ computable by a multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuit of size $s$, the algorithm runs in time ... more >>>

TR11-153 | 13th November 2011
Ankit Gupta, Neeraj Kayal, Satyanarayana V. Lokam

#### Reconstruction of Depth-4 Multilinear Circuits with Top fanin 2

We present a randomized algorithm for reconstructing multilinear depth-4 arithmetic circuits with fan-in 2 at the top + gate. The algorithm is given blackbox access to a multilinear polynomial f in F[x_1,..,x_n] computable by a multilinear Sum-Product-Sum-Product(SPSP) circuit of size s and outputs an equivalent multilinear SPSP circuit, runs ... more >>>

TR17-021 | 11th February 2017
Neeraj Kayal, Vineet Nair, Chandan Saha, Sébastien Tavenas

#### Reconstruction of full rank Algebraic Branching Programs

An algebraic branching program (ABP) A can be modelled as a product expression $X_1\cdot X_2\cdot \dots \cdot X_d$, where $X_1$ and $X_d$ are $1 \times w$ and $w \times 1$ matrices respectively, and every other $X_k$ is a $w \times w$ matrix; the entries of these matrices are linear forms ... more >>>

TR18-191 | 10th November 2018
Neeraj Kayal, Chandan Saha

#### Reconstruction of non-degenerate homogeneous depth three circuits

A homogeneous depth three circuit $C$ computes a polynomial
$$f = T_1 + T_2 + ... + T_s ,$$ where each $T_i$ is a product of $d$ linear forms in $n$ variables over some underlying field $\mathbb{F}$. Given black-box access to $f$, can we efficiently reconstruct (i.e. proper learn) a ... more >>>

TR14-147 | 6th November 2014
Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, David Zuckerman

Revisions: 1

We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$ where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently hard'' two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ ... more >>> TR01-004 | 13th October 2000 Tobias Gärtner, Günter Hotz #### Recursive analytic functions of a complex variable We extend the concept of recursive definition on analytic functions. For special cases of linear primitive recursive definitions we show the existence of natural continuations of the over$\N$primitive recursive functions to analytic functions. Especially, we show that solutions exist if the coefficients of the linear recursive equation are ... more >>> TR98-007 | 12th January 1998 Luca Trevisan #### Recycling Queries in PCPs and in Linearity Tests We study query-efficient Probabilistically Checkable Proofs (PCPs) and linearity tests. We focus on the number of amortized query bits. A testing algorithm uses$q$amortized query bits if, for some constant$k$, it reads$qk$bits and has error probability at most$2^{-k}$. The best known ... more >>> TR05-090 | 17th August 2005 Paul Goldberg, Christos H. Papadimitriou #### Reducibility Among Equilibrium Problems We address the fundamental question of whether the Nash equilibria of a game can be computed in polynomial time. We describe certain efficient reductions between this problem for normal form games with a fixed number of players and graphical games with fixed degree. Our main result is that ... more >>> TR09-041 | 9th April 2009 Shiva Kintali, Laura J Poplawski, Rajmohan Rajaraman, Ravi Sundaram, Shang-Hua Teng #### Reducibility Among Fractional Stability Problems "As has often been the case with NP-completeness proofs, PPAD-completeness proofs will be eventually refined to cover simpler and more realistic looking classes of games. And then researchers will strive to identify even simpler classes." --Papadimitriou (chapter 2 of Algorithmic Game Theory book) In a landmark paper, Papadimitriou introduced a ... more >>> TR11-113 | 11th August 2011 Emanuele Viola #### Reducing 3XOR to listing triangles, an exposition The 3SUM problem asks if there are three integers$a,b,c$summing to$0$in a given set of$n$integers of magnitude poly($n$). Patrascu (STOC '10) reduces solving 3SUM in time$n^{2-\Omega(1)}$to listing$m$triangles in a graph with$m$edges in time$m^{4/3-\Omega(1)}$. In this note we present ... more >>> TR99-018 | 8th June 1999 Manindra Agrawal, Somenath Biswas #### Reducing Randomness via Chinese Remaindering We give new randomized algorithms for testing multivariate polynomial identities over finite fields and rationals. The algorithms use \lceil \sum_{i=1}^n \log(d_i+1)\rceil (plus \lceil\log\log C\rceil in case of rationals where C is the largest coefficient) random bits to test if a polynomial P(x_1, ..., x_n) is zero where d_i is ... more >>> TR16-080 | 18th May 2016 Oded Goldreich #### Reducing testing affine spaces to testing linearity Revisions: 3 We consider the task of testing whether a Boolean function$f:\{0,1\}^\ell\to\{0,1\}$is the indicator function of an$(\ell-k)$-dimensional affine space. An optimal tester for this property was presented by Parnas, Ron, and Samorodnitsky ({\em SIDMA}, 2002), by mimicking the celebrated linearity tester (of Blum, Luby and Rubinfeld, {\em JCSS}, 1993) ... more >>> TR04-077 | 17th July 2004 Alina Beygelzimer, Varsha Dani, Tom Hayes, John Langford #### Reductions Between Classification Tasks There are two approaches to solving a new supervised learning task: either analyze the task independently or reduce it to a task that has already been thoroughly analyzed. This paper investigates the latter approach for classification problems. In addition to obvious theoretical motivations, there is fairly strong empirical evidence that ... more >>> TR03-027 | 21st April 2003 Christian Glaßer, Alan L. Selman, Samik Sengupta #### Reductions between Disjoint NP-Pairs We prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible ... more >>> TR10-172 | 11th November 2010 Prasad Raghavendra, David Steurer, Madhur Tulsiani #### Reductions Between Expansion Problems The Small-Set Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the Unique Games Conjecture (Khot, STOC 2002). In particular, the Small-Set Expansion Hypothesis implies the Unique ... more >>> TR07-071 | 1st August 2007 Jacobo Toran #### Reductions to Graph Isomorphism We show that several reducibility notions coincide when applied to the Graph Isomorphism (GI) problem. In particular we show that if a set is many-one logspace reducible to GI, then it is in fact many-one AC^0 reducible to GI. For the case of Turing reducibilities we show that ... more >>> TR12-054 | 2nd May 2012 Eric Allender, Harry Buhrman, Luke Friedman, Bruno Loff #### Reductions to the set of random strings:the resource-bounded case Revisions: 1 This paper is motivated by a conjecture that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out by [Allender et al] to settle this conjecture cannot succeed without significant alteration, but that it ... more >>> TR05-068 | 7th July 2005 Christian Glaßer, A. Pavan, Alan L. Selman, Liyu Zhang #### Redundancy in Complete Sets We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glasser et al., complete sets for all of the following complexity classes are m-mitotic: ... more >>> TR10-052 | 8th March 2010 Melanie Winkler, Berthold Vöcking, Sascha Geulen #### Regret Minimization for Online Buffering Problems Using the Weighted Majority Algorithm Suppose a decision maker has to purchase a commodity over time with varying prices and demands. In particular, the price per unit might depend on the amount purchased and this price function might vary from step to step. The decision maker has a buffer of bounded size for storing units ... more >>> TR11-173 | 22nd December 2011 Christoph Behle, Andreas Krebs #### Regular Languages in MAJ[<] with three variables We consider first order logic over words and show FO+MOD[<] is contained in MAJ[<] with three variables. It is known that for the classes FO[<], FO+MOD[<], FO+GROUP[<] three variables suffice. In the case of MOD[<] even two variables are sufficient. As a consequence we know that if TC^ 0 neq ... more >>> TR08-103 | 22nd November 2008 Luca Trevisan, Madhur Tulsiani, Salil Vadhan #### Regularity, Boosting, and Efficiently Simulating Every High-Entropy Distribution We show that every high-entropy distribution is indistinguishable from an efficiently samplable distribution of the same entropy. Specifically, we prove that if$D$is a distribution over$\{ 0,1\}^n$of min-entropy at least$n-k$, then for every$S$and$\epsilon$there is a circuit$C$of size at most$S\cdot ... more >>>

TR16-164 | 25th October 2016
Andreas Krebs, Meena Mahajan, Anil Shukla

#### Relating two width measures for resolution proofs

In this short note, we revisit two hardness measures for resolution proofs: width and asymmetric width. It is known that for every unsatisfiable CNF F,

width(F \derives \Box) \le awidth(F \derives \Box) + max{ awidth(F \derives \Box), width(F)}.

We give a simple direct proof of the upper bound, ... more >>>

TR10-040 | 10th March 2010
Pavel Hrubes, Avi Wigderson, Amir Yehudayoff

#### Relationless completeness and separations

This paper extends Valiant's work on $\vp$ and $\vnp$ to the settings in which variables are not multiplicatively commutative and/or associative. Our main result is a theory of completeness for these algebraic worlds.
We define analogs of Valiant's classes $\vp$ and $\vnp$, as well as of the polynomials permanent ... 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 >>>

TR15-028 | 27th February 2015
Lila Fontes, Rahul Jain, Iordanis Kerenidis, Sophie Laplante, Mathieu Laurière, Jérémie Roland

#### Relative Discrepancy does not separate Information and Communication Complexity

Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Recently, Ganor et al. provided such a separation in the distributional setting for a specific input distribution ?. We show that ... more >>>

TR18-103 | 30th April 2018
Zhao Song, David Woodruff, Peilin Zhong

#### Relative Error Tensor Low Rank Approximation

We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-$q$ tensor $A \in \mathbb{R}^{\prod_{i=1}^q n_i}$, output a rank-$k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+\epsilon) {\rm OPT}$, where ${\rm OPT} = \inf_{\textrm{rank-}k~A'} \|A-A'\|_F^2$. Despite much success on obtaining relative error low ... more >>>

TR01-068 | 19th September 2001
Philippe Moser

#### Relative to P, APP and promise-BPP are the same

Revisions: 1

$\mathbf{APP}$, the class of real functions
approximable by probabilistic Turing machines, is the same as having oracle access to
promise-$\mathbf{BPP}$. First
we construct a mapping that maps every function in $\mathbf{APP}$ to a promise problem
more >>>

TR03-084 | 27th November 2003
Joshua Buresh-Oppenheim, Tsuyoshi Morioka

#### Relativized NP Search Problems and Propositional Proof Systems

We consider Total Functional $\NP$ ($\TFNP$) search problems. Such problems are based on combinatorial principles that guarantee, through locally checkable conditions, that a solution to the problem exists in an exponentially-large domain, and have the property that any solution has a polynomial-size witness that can be verified in polynomial time. ... more >>>

TR10-042 | 12th March 2010
Thomas Watson

#### Relativized Worlds Without Worst-Case to Average-Case Reductions for NP

Revisions: 3

We prove that relative to an oracle, there is no worst-case to errorless-average-case reduction for $\NP$. This result is the first progress on an open problem posed by Impagliazzo in 1995, namely to construct an oracle relative to which $\NP$ is worst-case hard but errorless-average-case easy. We also handle classes ... more >>>

TR17-143 | 26th September 2017
Tom Gur, Govind Ramnarayan, Ron Rothblum

#### Relaxed Locally Correctable Codes

Revisions: 1

Locally decodable codes (LDCs) and locally correctable codes (LCCs) are error-correcting codes in which individual bits of the message and codeword, respectively, can be recovered by querying only few bits from a noisy codeword. These codes have found numerous applications both in theory and in practice.

A natural relaxation of ... more >>>

TR15-199 | 7th December 2015

#### Relaxed partition bound is quadratically tight for product distributions

Let $f : \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ be a 2-party function. For every product distribution $\mu$ on $\{0,1\}^n \times \{0,1\}^n$, we show that $${{CC}}^\mu_{0.49}(f) = O\left(\left(\log {{rprt}}_{1/4}(f) \cdot \log \log {{rprt}}_{1/4}(f)\right)^2\right),$$ where ${{CC}^\mu_\varepsilon(f)$ is the distributional communication complexity with error at most $\varepsilon$ under the distribution $\mu$ and ... more >>>

TR15-014 | 18th January 2015
Noga Alon, Mark Braverman, Klim Efremenko, Ran Gelles, Bernhard Haeupler

#### Reliable Communication over Highly Connected Noisy Networks

We consider the task of multiparty computation performed over networks in
the presence of random noise. Given an $n$-party protocol that takes $R$
rounds assuming noiseless communication, the goal is to find a coding
scheme that takes $R'$ rounds and computes the same function with high
probability even when the ... more >>>

TR18-041 | 26th February 2018
Sam Buss, Dmitry Itsykson, Alexander Knop, Dmitry Sokolov

#### Reordering Rule Makes OBDD Proof Systems Stronger

Atserias, Kolaitis, and Vardi [AKV04] showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD($\land$, weakening), simulates CP* (Cutting Planes with unary coefficients). We show that OBDD($\land$, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring ... more >>>

TR04-082 | 9th September 2004
Olaf Beyersdorff

#### Representable Disjoint NP-Pairs

Revisions: 1

We investigate the class of disjoint NP-pairs under different reductions.
The structure of this class is intimately linked to the simulation order
of propositional proof systems, and we make use of the relationship between
propositional proof systems and theories of bounded arithmetic as the main
tool of our analysis.
more >>>

TR15-157 | 1st September 2015
Thomas O'Neil

#### Representation-Independent Fixed Parameter Tractability for Vertex Cover and Weighted Monotone Satisfiability

A symmetric representation for a set of objects requires the same amount of space for the set as for its complement. Complexity classifications that are based on the length of the input can depend on whether the representation is symmetric. In this article we describe a symmetric representation scheme for ... more >>>

TR17-106 | 16th June 2017
Mateus de Oliveira Oliveira, Pavel Pudlak

#### Representations of Monotone Boolean Functions by Linear Programs

We introduce the notion of monotone linear programming circuits (MLP circuits), a model of
computation for partial Boolean functions. Using this model, we prove the following results:

1. MLP circuits are superpolynomially stronger than monotone Boolean circuits.
2. MLP circuits are exponentially stronger than monotone span programs.
3. ... more >>>

TR06-158 | 8th December 2006
Gyula Gyôr

#### Representing Boolean OR function by quadratic polynomials modulo 6

We give an answer to the question of Barrington, Beigel and Rudich, asked in 1992, concerning the largest n such that the OR function of n variable can be weakly represented by a quadratic polynomial modulo 6. More specially,we show that no 11-variable quadratic polynomial exists that is congruent to ... more >>>

TR18-048 | 11th March 2018
Ofer Grossman, Yang P. Liu

#### Reproducibility and Pseudo-Determinism in Log-Space

A curious property of randomized log-space search algorithms is that their outputs are often longer than their workspace. This leads to the question: how can we reproduce the results of a randomized log space computation without storing the output or randomness verbatim? Running the algorithm again with new random bits ... more >>>

TR99-042 | 24th October 1999
Ran Canetti, Oded Goldreich, Silvio Micali.

#### Resettable Zero-Knowledge.

Revisions: 1

We introduce the notion of Resettable Zero-Knowledge (rZK),
a new security measure for cryptographic protocols
which strengthens the classical notion of zero-knowledge.
In essence, an rZK protocol is one that remains zero knowledge
even if an adeversary can interact with the prover many times, each
time ... more >>>

TR04-112 | 26th November 2004
Neil Thapen, Nicola Galesi

#### Resolution and pebbling games

We define a collection of Prover-Delayer games that characterize certain subsystems of resolution. This allows us to give some natural criteria which guarantee lower bounds on the resolution width of a formula, and to extend these results to formulas of unbounded initial width.

We also use games to give upper ... more >>>

TR18-165 | 20th September 2018
Stefan Dantchev, Nicola Galesi, Barnaby Martin

#### Resolution and the binary encoding of combinatorial principles

We investigate the size complexity of proofs in $RES(s)$ -- an extension of Resolution working on $s$-DNFs instead of clauses -- for families of contradictions given in the {\em unusual binary} encoding. A motivation of our work is size lower bounds of refutations in Resolution for families of contradictions in ... more >>>

TR14-093 | 22nd July 2014
Dmitry Itsykson, Mikhail Slabodkin, Dmitry Sokolov

#### Resolution complexity of perfect mathcing principles for sparse graphs

The resolution complexity of the perfect matching principle was studied by Razborov [Raz04], who developed a technique for proving its lower bounds for dense graphs. We construct a constant degree bipartite graph $G_n$ such that the resolution complexity of the perfect matching principle for $G_n$ is $2^{\Omega(n)}$, where $n$ is ... 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 >>>

TR01-075 | 2nd November 2001
Alexander Razborov

#### Resolution Lower Bounds for the Weak Functional Pigeonhole Principle

We show that every resolution proof of the {\em functional} version
$FPHP^m_n$ of the pigeonhole principle (in which one pigeon may not split
between several holes) must have size $\exp\of{\Omega\of{\frac n{(\log m)^2}}}$. This implies an $\exp\of{\Omega(n^{1/3})}$ bound when the number
of pigeons $m$ is arbitrary.

more >>>

TR01-021 | 7th March 2001
Ran Raz

#### Resolution Lower Bounds for the Weak Pigeonhole Principle

Revisions: 1

We prove that any Resolution proof for the weak
pigeon hole principle, with $n$ holes and any number of
pigeons, is of length $\Omega(2^{n^{\epsilon}})$,
(for some global constant $\epsilon > 0$).

more >>>

TR07-078 | 11th August 2007
Ran Raz, Iddo Tzameret

#### Resolution over Linear Equations and Multilinear Proofs

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. ... more >>>

TR18-117 | 23rd June 2018
Fedor Part, Iddo Tzameret

#### Resolution with Counting: Lower Bounds over Different Moduli

Revisions: 1

Resolution over linear equations (introduced in [RT08]) emerged recently as an important object of study. This refutation system, denoted Res(lin$_R$), operates with disjunction of linear equations over a ring $R$. On the one hand, the system captures a natural minimal'' extension of resolution in which efficient counting can be achieved; ... more >>>

TR01-085 | 1st October 2001
Gerhard J. Woeginger

#### Resource augmentation for online bounded space bin packing

We study online bounded space bin packing in the resource
augmentation model of competitive analysis.
In this model, the online bounded space packing algorithm has
to pack a list L of items in (0,1] into a small number of
bins of size b>=1.
Its performance is measured by comparing the ... more >>>

TR04-066 | 6th July 2004
Tomoyuki Yamakami, Toshio Suzuki

#### Resource Bounded Immunity and Simplicity

Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we ... more >>>

TR02-038 | 5th June 2002
Rahul Santhanam

Revisions: 1

We consider uniform assumptions for derandomization. We provide
intuitive evidence that BPP can be simulated non-trivially in
deterministic time by showing that (1) P \not \subseteq i.o.i.PLOYLOGSPACE
implies BPP \subseteq SUBEXP (2) P \not \subseteq SUBPSPACE implies BPP
= P. These results extend and complement earlier work of ... more >>>

TR17-119 | 25th July 2017

#### Resource-Efficient Common Randomness and Secret-Key Schemes

We study common randomness where two parties have access to i.i.d. samples from a known random source, and wish to generate a shared random key using limited (or no) communication with the largest possible probability of agreement. This problem is at the core of secret key generation in cryptography, with ... more >>>

TR12-082 | 28th June 2012
Mahdi Cheraghchi, Venkatesan Guruswami, Ameya Velingker

#### Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

We prove that a random linear code over $\mathbb{F}_q$, with probability arbitrarily close to $1$, is list decodable at radius $1-1/q-\epsilon$ with list size $L=O(1/\epsilon^2)$ and rate $R=\Omega_q(\epsilon^2/(\log^3(1/\epsilon)))$. Up to the polylogarithmic factor in $1/\epsilon$ and constant factors depending on $q$, this matches the lower bound $L=\Omega_q(1/\epsilon^2)$ for the list ... more >>>

TR00-048 | 3rd July 2000
Beate Bollig

#### Restricted Nondeterministic Read-Once Branching Programs and an Exponential Lower Bound for Integer Multiplication

Branching programs are a well established computation model for
Boolean functions, especially read-once branching programs have
been studied intensively.
In this paper the expressive power of nondeterministic read-once
branching programs, i.e., the class of functions
representable in polynomial size, is investigated.
For that reason two restricted models of nondeterministic read-once
more >>>

TR09-094 | 7th October 2009
Bireswar Das, Jacobo Toran, Fabian Wagner

#### Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs

The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width
are known to be solvable in polynomial time \cite{Bo90},\cite{YBFT}.
We give restricted space algorithms for these problems proving the following results:

Isomorphism for bounded tree distance width graphs is in L and thus complete ... more >>>

TR11-160 | 1st December 2011
Zeev Dvir, Anup Rao, Avi Wigderson, Amir Yehudayoff

#### Restriction Access

We introduce a notion of non-black-box access to computational devices (such as circuits, formulas, decision trees, and so forth) that we call \emph{restriction access}. Restrictions are partial assignments to input variables. Each restriction simplifies the device, and yields a new device for the restricted function on the unassigned variables. On ... more >>>

TR19-097 | 4th July 2019
Jacobo Toran, Florian Wörz

#### Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space

We show a new connection between the space measure in tree-like resolution and the reversible pebble game in graphs. Using this connection we provide several formula classes for which there is a logarithmic factor separation between the space complexity measure in tree-like and general resolution. We show that these separations ... more >>>

TR05-032 | 16th March 2005
Gudmund Skovbjerg Frandsen, Peter Bro Miltersen

#### Reviewing Bounds on the Circuit Size of the Hardest Functions

In this paper we review the known bounds for $L(n)$, the circuit size
complexity of the hardest Boolean function on $n$ input bits. The
best known bounds appear to be $$\frac{2^n}{n}(1+\frac{\log n}{n}-O(\frac{1}{n})) \leq L(n) \leq\frac{2^n}{n}(1+3\frac{\log n}{n}+O(\frac{1}{n}))$$ However, the bounds do not seem to be
explicitly stated in the literature. We ... more >>>

TR19-092 | 9th July 2019
Venkatesan Guruswami, Jakub Opršal, Sai Sandeep

#### Revisiting Alphabet Reduction in Dinur's PCP

Dinur's celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the ... more >>>

TR13-113 | 19th August 2013
Moritz Müller, Stefan Szeider

#### Revisiting Space in Proof Complexity: Treewidth and Pathwidth

So-called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs. The ordered pathwidth of a proof is shown to be roughly the same as its formula space. Length-space lower bounds for R(k)-refutations ... more >>>

TR02-043 | 11th July 2002
Dalit Naor, Moni Naor, Jeff Lotspiech

#### Revocation and Tracing Schemes for Stateless Receivers

We deal with the problem of a center sending a secret message to
a group of users such that some subset of the users is considered
revoked and should not be able to obtain the content of the
message. We concentrate on the stateless receiver case, where
the users do ... more >>>

TR14-066 | 17th April 2014
Suguru Tamaki, Yuichi Yoshida

#### Robust Approximation of Temporal CSP

A temporal constraint language $\Gamma$ is a set of relations with first-order definitions in $({\mathbb{Q}}; <)$. Let CSP($\Gamma$) denote the set of constraint satisfaction problem instances with relations from $\Gamma$. CSP($\Gamma$) admits robust approximation if, for any $\varepsilon \geq 0$, given a $(1-\varepsilon)$-satisfiable instance of CSP($\Gamma$), we can compute an ... more >>>

TR06-118 | 2nd September 2006
Irit Dinur, Madhu Sudan, Avi Wigderson

#### Robust Local Testability of Tensor Products of LDPC Codes

Given two binary linear codes R and C, their tensor product R \otimes C consists of all matrices with rows in R and columns in C. We analyze the "robustness" of the following test for this code (suggested by Ben-Sasson and Sudan~\cite{BenSasson-Sudan04}): Pick a random row (or column) and check ... more >>>

TR04-046 | 4th June 2004

#### Robust Locally Testable Codes and Products of Codes

We continue the investigation of locally testable codes, i.e.,
error-correcting codes for whom membership of a given word in the
code can be tested probabilistically by examining it in very few
locations. We give two general results on local testability:
First, motivated by the recently proposed notion of robust
probabilistically ... more >>>

TR11-062 | 18th April 2011
Amit Chakrabarti, Graham Cormode, Andrew McGregor

#### Robust Lower Bounds for Communication and Stream Computation

We study the communication complexity of evaluating functions when the input data is randomly allocated (according to some known distribution) amongst two or more players, possibly with information overlap. This naturally extends previously studied variable partition models such as the best-case and worst-case partition models. We aim to understand whether ... more >>>

TR16-204 | 20th December 2016

#### Robust Multiplication-based Tests for Reed-Muller Codes

We consider the following multiplication-based tests to check if a given function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is the evaluation of a degree-$d$ polynomial over $\mathbb{F}_q$ for $q$ prime.

* $\mathrm{Test}_{e,k}$: Pick $P_1,\ldots,P_k$ independent random degree-$e$ polynomials and accept iff the function $fP_1\cdots P_k$ is the evaluation of a degree-$(d+ek)$ polynomial.

... more >>>

TR04-021 | 23rd March 2004

#### Robust PCPs of Proximity, Shorter PCPs and Applications to Coding

We continue the study of the trade-off between the length of PCPs
and their query complexity, establishing the following main results
(which refer to proofs of satisfiability of circuits of size $n$):
We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$
that can be verified by making $o(\log\log n)$ Boolean queries.
more >>>

TR13-143 | 19th October 2013
Yuval Ishai, Eyal Kushilevitz, Xin Li, Rafail Ostrovsky, Manoj Prabhakaran, Amit Sahai, David Zuckerman

#### Robust Pseudorandom Generators

Revisions: 1

Let $G:\{0,1\}^n\to\{0,1\}^m$ be a pseudorandom generator. We say that a circuit implementation of $G$ is $(k,q)$-robust if for every set $S$ of at most $k$ wires anywhere in the circuit, there is a set $T$ of at most $q|S|$ outputs, such that conditioned on the values of $S$ and $T$ ... more >>>

TR11-163 | 2nd December 2011
Libor Barto, Marcin Kozik

#### Robust Satisfiability of Constraint Satisfaction Problems

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least $(1-g(\varepsilon))$-fraction of the constraints given a $(1-\varepsilon)$-satisfiable instance, where $g(\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$, $g(0)=0$.
Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction ... more >>>

TR16-161 | 26th October 2016
Shachar Lovett, Jiapeng Zhang

#### Robust sensitivity

Revisions: 1

The sensitivity conjecture is one of the central open problems in boolean complexity. A recent work of Gopalan et al. [CCC 2016] conjectured a robust analog of the sensitivity conjecture, which relates the decay of the Fourier mass of a boolean function to moments of its sensitivity. We prove this ... more >>>

TR15-043 | 2nd April 2015

#### Robust testing of lifted codes with applications to low-degree testing

A local tester for a code probabilistically looks at a given word at a small set of coordinates and based on this local view accepts codewords with probability one while rejecting words far from the code with constant probabilility. A local tester for a code is said to be robust'' ... more >>>

TR17-055 | 26th March 2017
Maya Leshkowitz

#### Round Complexity Versus Randomness Complexity in Interactive Proofs

Consider an interactive proof system for some set S that has randomness complexity r(n) for instances of length n, and arbitrary round complexity. We show a public-coin interactive proof system for S of round complexity O(r(n)/log n). Furthermore, the randomness complexity is preserved up to a constant factor, and the ... more >>>

TR06-093 | 27th July 2006
Takeshi Koshiba, Yoshiharu Seri

#### Round-Efficient One-Way Permutation Based Perfectly Concealing Bit Commitment Scheme

We explicitly show the upper bound on the round complexity for perfectly concealing bit commitment schemes based on the general computational assumption. The best known scheme in the literature is the one-way permutation based scheme due to Naor, Ostrovsky, Venkatesan and Yung and its round complexity is O(n). We consider ... more >>>

TR11-065 | 25th April 2011
Boaz Barak, Prasad Raghavendra, David Steurer

#### Rounding Semidefinite Programming Hierarchies via Global Correlation

We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with ... more >>>

TR13-184 | 23rd December 2013
Boaz Barak, Jonathan Kelner, David Steurer

#### Rounding Sum-of-Squares Relaxations

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a *combining algorithm* -- an algorithm that maps a distribution over solutions into a (possibly ... more >>>

TR03-040 | 3rd June 2003
Philippe Moser

#### RP is Small in SUBEXP else ZPP equals PSPACE and NP equals EXP

We use recent results on the hardness of resource-bounded
Kolmogorov random strings, to prove that RP is small in SUBEXP
else ZPP=PSPACE and NP=EXP.
We also prove that if NP is not small in SUBEXP, then
NP=AM, improving a former result which held for the measure ... more >>>

TR06-084 | 19th June 2006
Frank Neumann, Carsten Witt

#### Runtime Analysis of a Simple Ant Colony Optimization Algorithm

Ant Colony Optimization (ACO) has become quite popular in recent
years. In contrast to many successful applications, the theoretical
foundation of this randomized search heuristic is rather weak.
Building up such a theory is demanded to understand how these
heuristics work as well as to ... more >>>

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