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TR17-004 | 8th January 2017
Scott Aaronson

P=?NP

In 1955, John Nash sent a remarkable letter to the National Security Agency, in which—seeking to build theoretical foundations for cryptography—he all but formulated what today we call the P=?NP problem, considered one of the great open problems of science. Here I survey the status of this problem in 2017, ... more >>>


TR12-013 | 15th February 2012
Tomas Feder, Carlos Subi

Packing Edge-Disjoint Triangles in Given Graphs

Given a graph $G$, we consider the problem of finding the largest set
of edge-disjoint triangles contained in $G$. We show that even the
simpler case of decomposing the edges of
a sparse split graph $G$ into edge-disjoint triangles
is NP-complete. We show next that the case of a general ... more >>>


TR06-030 | 17th January 2006
Piotr Berman, Jieun Jeong, Shiva Prasad Kasiviswanathan, Bhuvan Urgaonkar

Packing to angles and sectors

In our problem we are given a set of customers, their positions on the
plane and their demands. Geometrically, directional antenna with
parameters $\alpha,\rho,R$ is a set
of points with radial coordinates $(\theta,r)$ such that
$\alpha \le \theta \le \alpha+\rho$ and $r \le R$. Given a set of
possible directional ... more >>>


TR20-014 | 16th February 2020
Gil Cohen, Shahar Samocha

Palette-Alternating Tree Codes

A tree code is an edge-coloring of the complete infinite binary tree such that every two nodes of equal depth have a fraction--bounded away from $0$--of mismatched colors between the corresponding paths to their least common ancestor. Tree codes were introduced in a seminal work by Schulman (STOC 1993) and ... more >>>


TR12-066 | 22nd April 2012
Jinyu Huang

Parallel Complexity for Matroid Intersection and Matroid Parity Problems

Revisions: 1

Let two linear matroids have the same rank in matroid intersection.
A maximum linear matroid intersection (maximum linear matroid parity
set) is called a basic matroid intersection (basic matroid parity
set), if its size is the rank of the matroid. We present that
enumerating all basic matroid intersections (basic matroid ... more >>>


TR98-009 | 25th November 1997
Bruce Edward Litow

Parallel complexity of integer coprimality

Comments: 3

It is shown that integer coprimality is in NC.

more >>>

TR02-029 | 3rd June 2002
Marek Karpinski, Yakov Nekrich

Parallel Construction of Minimum Redundancy Length-Limited Codes

This paper presents new results on parallel constructions of the
length-limited prefix-free codes with the minimum redundancy.
We describe an algorithm for the construction of length-limited codes
that works in $O(L)$ time with $n$ processors for $L$ the
maximal codeword length.
We also describe an algorithm for a construction ... more >>>


TR00-053 | 5th May 2000
Alexander E. Andreev, Andrea E. F. Clementi, Paolo Penna, Jose' D. P. Rolim

Parallel Read Operations Without Memory Contention

We address the problem of organizing a set $T$ of shared data into
the memory modules of a Distributed Memory Machine (DMM) in order
to minimize memory access conflicts (i.e. memory contention)
during read operations.
Previous solutions for this problem can be found as fundamental ... more >>>


TR22-039 | 14th March 2022
Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, Wei Zhan

Parallel Repetition For All 3-Player Games Over Binary Alphabet

We prove that for every 3-player (3-prover) game, with binary questions and answers and value less than $1$, the value of the $n$-fold parallel repetition of the game decays polynomially fast to $0$. That is, for every such game, there exists a constant $c>0$, such that the value of the ... more >>>


TR22-167 | 23rd November 2022
Mark Braverman, Subhash Khot, Dor Minzer

Parallel Repetition for the GHZ Game: Exponential Decay

We show that the value of the $n$-fold repeated GHZ game is at most $2^{-\Omega(n)}$, improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup type questions from additive combinatorics.

more >>>

TR08-013 | 16th January 2008
Anup Rao

Parallel Repetition in Projection Games and a Concentration Bound

In a two player game, a referee asks two cooperating players (who are
not allowed to communicate) questions sampled from some distribution
and decides whether they win or not based on some predicate of the
questions and their answers. The parallel repetition of the game is
the game in which ... more >>>


TR14-054 | 16th April 2014
Dana Moshkovitz

Parallel Repetition of Fortified Games

Revisions: 2

The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game G^k in terms of the value of the game G and the number of repetitions k.
Contrary to what one might have guessed, the value of G^k is not val(G)^k, but rather a more complicated ... more >>>


TR23-198 | 8th December 2023
Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, Dor Minzer

Parallel Repetition of k-Player Projection Games

We study parallel repetition of k-player games where the constraints satisfy the projection property. We prove exponential decay in the value of a parallel repetition of projection games with value less than 1.

more >>>

TR16-047 | 23rd March 2016
Mohammad Bavarian, Thomas Vidick, Henry Yuen

Parallel repetition via fortification: analytic view and the quantum case

In a recent work, Moshkovitz [FOCS '14] presented a transformation on two-player games called "fortification'', and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial ... more >>>


TR10-195 | 13th November 2010
Ho-Lin Chen, David Doty, Shinnosuke Seki, David Soloveichik

Parallelism, Program Size, Time, and Temperature in Self-Assembly

Revisions: 1

We settle a number of questions in variants of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the "hierarchical" aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the "seeded" aTAM, tiles attach one at a time to a ... more >>>


TR06-072 | 25th February 2006
Henning Fernau

Parameterized Algorithms for Hitting Set: the Weighted Case

We are going to analyze simple search tree algorithms
for Weighted d-Hitting Set. Although the algorithms are simple, their analysis is technically rather involved. However, this approach allows us to even improve on elsewhere published algorithm running time estimates for the more restricted case of (unweighted) d-Hitting Set.

... more >>>

TR10-198 | 13th December 2010
Olaf Beyersdorff, Nicola Galesi, Massimo Lauria, Alexander Razborov

Parameterized Bounded-Depth Frege is Not Optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that framework the parameterized version of any proof system is not fpt-bounded for some technical reasons, but we remark that this question becomes much more interesting if we restrict ourselves to those parameterized contradictions ... more >>>


TR14-134 | 10th October 2014
Martin Lück, Arne Meier, Irina Schindler

Parameterized Complexity of CTL: Courcelle's Theorem For Infinite Vocabularies

Revisions: 3

We present a complete classification of the parameterized complexity of all operator fragments of the satisfiability problem in computation tree logic CTL. The investigated parameterization is temporal depth and pathwidth. Our results show a dichotomy between W[1]-hard and fixed-parameter tractable fragments. The two real operator fragments which are in FPT ... more >>>


TR09-131 | 2nd December 2009
Stephan Kreutzer, Anuj Dawar

Parameterized Complexity of First-Order Logic

Revisions: 2

We show that if $\mathcal C$ is a class of graphs which is
"nowhere dense" then first-order model-checking is
fixed-parameter tractable on $\mathcal C$. As all graph classes which exclude a fixed minor, or are of bounded local tree-width or locally exclude a minor are nowhere dense, this generalises algorithmic ... more >>>


TR16-157 | 13th October 2016
Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, Jacobo Toran

Parameterized Complexity of Small Weight Automorphisms

We consider the PermCode problem to decide, given a representation of a permutation group G and a parameter k, whether there is a non-trivial element of G with support at most k. This problem generalizes several problems in the literature. We introduce a new method that allows to reduce the ... more >>>


TR01-023 | 13th March 2001
Jochen Alber, Henning Fernau, Rolf Niedermeier

Parameterized Complexity: Exponential Speed-Up for Planar Graph Problems

Revisions: 1

A parameterized problem is called fixed parameter tractable
if it admits a solving algorithm whose running time on
input instance $(I,k)$ is $f(k) \cdot |I|^\alpha$, where $f$
is an arbitrary function depending only on~$k$. Typically,
$f$ is some exponential function, e.g., $f(k)=c^k$ for ... more >>>


TR23-188 | 28th November 2023
Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, Kewen Wu

Parameterized Inapproximability Hypothesis under ETH

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. PIH plays the role of ... more >>>


TR22-156 | 15th November 2022
Huck Bennett, Mahdi Cheraghchi, Venkatesan Guruswami, Joao Ribeiro

Parameterized Inapproximability of the Minimum Distance Problem over all Fields and the Shortest Vector Problem in all $\ell_p$ Norms

Revisions: 2

We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that ... more >>>


TR19-115 | 4th September 2019
Arnab Bhattacharyya, Édouard Bonnet, László Egri, Suprovat Ghoshal, Karthik C. S., Bingkai Lin, Pasin Manurangsi, Dániel Marx

Parameterized Intractability of Even Set and Shortest Vector Problem

The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over $\mathbb{F}_2$, which can be stated as follows: given a generator matrix A and an integer k, determine whether the code generated by A has distance at most k, or in other words, whether ... more >>>


TR18-057 | 26th March 2018
Arnab Bhattacharyya, Suprovat Ghoshal, Karthik C. S., Pasin Manurangsi

Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH

The $k$-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over $\mathbb F_2$, which can be stated as follows: given a generator matrix $\mathbf A$ and an integer $k$, determine whether the code generated by $\mathbf A$ has distance at most $k$. Here, $k$ ... more >>>


TR97-006 | 31st January 1997
Marco Cesati, Miriam Di Ianni

Parameterized Parallel Complexity

Comments: 1

We consider the framework of Parameterized Complexity, and we
investigate the issue of which problems do admit efficient fixed
parameter parallel algorithms by introducing two different
degrees of efficiently parallelizable parameterized problems, PNC
and FPP. We sketch both some FPP-algorithms solving natural
parameterized problems and ... more >>>


TR07-001 | 19th November 2006
Stefan S. Dantchev, Barnaby Martin, Stefan Szeider

Parameterized Proof Complexity: a Complexity Gap for Parameterized Tree-like Resolution

Revisions: 1

We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter. One could separate the ... more >>>


TR17-103 | 12th June 2017
Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, Nithin Varma

Parameterized Property Testing of Functions

We investigate the parameters in terms of which the complexity of sublinear-time algorithms should be expressed. Our goal is to find input parameters that are tailored to the combinatorics of the specific problem being studied and design algorithms that run faster when these parameters are small. This direction enables us ... more >>>


TR04-027 | 20th February 2004
Henning Fernau

Parametric Duality: Kernel Sizes and Algorithmics

We derive the first lower bound results on kernel sizes of parameterized problems. The same idea also allows us to sometimes "de-parameterize"
parameterized algorithms.

more >>>

TR18-211 | 3rd December 2018
Kshitij Gajjar, Jaikumar Radhakrishnan

Parametric Shortest Paths in Planar Graphs

Revisions: 1

We construct a family of planar graphs $(G_n: n\geq 4)$, where $G_n$ has $n$ vertices including a source vertex $s$ and a sink vertex $t$, and edge weights that change linearly with a parameter $\lambda$ such that, as $\lambda$ increases, the cost of the shortest path from $s$ to $t$ ... more >>>


TR97-033 | 1st August 1997
Meena Mahajan, Venkatesh Raman

Parametrizing Above Guaranteed Values: MaxSat and MaxCut

In this paper we investigate the parametrized
complexity of the problems MaxSat and MaxCut using the
framework developed by Downey and Fellows.

Let $G$ be an arbitrary graph having $n$ vertices and $m$
edges, and let $f$ be an arbitrary CNF formula with $m$
clauses on $n$ variables. We improve ... more >>>


TR10-153 | 7th October 2010
Lorenzo Carlucci, Nicola Galesi, Massimo Lauria

Paris-Harrington tautologies

Revisions: 2

We initiate the study of the proof complexity of propositional encoding of (weak cases of) concrete independence results. In particular we study the proof complexity of Paris-Harrington's Large Ramsey Theorem. We prove a conditional lower bound in Resolution and a quasipolynomial upper bound in bounded-depth Frege.

more >>>

TR18-174 | 19th October 2018
Anastasiya Chistopolskaya, Vladimir Podolskii

Parity Decision Tree Complexity is Greater Than Granularity

Revisions: 2

We prove a new lower bound on the parity decision tree complexity $D_{\oplus}(f)$ of a Boolean function $f$. Namely, granularity of the Boolean function $f$ is the smallest $k$ such that all Fourier coefficients of $f$ are integer multiples of $1/2^k$. We show that $D_{\oplus}(f)\geq k+1$.

This lower bound is ... more >>>


TR13-092 | 19th June 2013
Pavel Pudlak, Arnold Beckmann, Neil Thapen

Parity Games and Propositional Proofs

Revisions: 1

A propositional proof system is \emph{weakly automatizable} if there
is a polynomial time algorithm which separates satisfiable formulas
from formulas which have a short refutation in the system, with
respect to a given length bound. We show that if the resolution
proof system is weakly automatizable, ... more >>>


TR01-073 | 24th October 2001
Beate Bollig, Philipp Woelfel, Stephan Waack

Parity Graph-driven Read-Once Branching Programs and an Exponential Lower Bound for Integer Multiplication

Revisions: 1


Branching programs are a well-established computation model
for Boolean functions, especially read-once branching programs
have been studied intensively. Exponential lower bounds for
deterministic and nondeterministic read-once branching programs
are known for a long time. On the other hand, the problem of
proving superpolynomial lower bounds ... 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 >>>


TR07-035 | 3rd April 2007
Mark Braverman, Raghav Kulkarni, Sambuddha Roy

Parity Problems in Planar Graphs

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2^k, for constant k. ... more >>>


TR04-025 | 24th January 2004
John Hitchcock, A. Pavan, N. V. Vinodchandran

Partial Bi-Immunity and NP-Completeness

The Turing and many-one completeness notions for $\NP$ have been
previously separated under {\em measure}, {\em genericity}, and {\em
bi-immunity} hypotheses on NP. The proofs of all these results rely
on the existence of a language in NP with almost everywhere hardness.

In this paper we separate the same NP-completeness ... more >>>


TR14-011 | 22nd January 2014
Dmytro Gavinsky, Pavel Pudlak

Partition Expanders

We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders it is required that the number of edges between any ... more >>>


TR06-016 | 1st February 2006
Tomas Feder, Carlos Subi

Partition into $k$-vertex subgraphs of $k$-partite graphs

The $H$-matching problem asks to partition the vertices of an input graph $G$
into sets of size $k=|V(H)|$, each of which induces a subgraph of $G$
isomorphic to $H$. The $H$-matching problem has been classified as polynomial
or NP-complete depending on whether $k\leq 2$ or not. We consider a variant
more >>>


TR05-095 | 26th August 2005
Noga Alon, Ilan Newman, Alexander Shen, Gábor Tardos, Nikolay Vereshchagin

Partitioning multi-dimensional sets in a small number of ``uniform'' parts

Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log |E|) subsets so the all the resulting bipartite graphcs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of ... more >>>


TR05-002 | 6th January 2005
Magnus Bordewich, Martin Dyer, Marek Karpinski

Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs

We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree $\Delta$ of ... more >>>


TR95-022 | 2nd May 1995
Marek Karpinski, Wojciech Rytter, Ayumi Shinohara

Pattern-Matching for Strings with Short Descriptions

We consider strings which are succinctly described. The description
is in terms of straight-line programs in which the constants are
symbols and the only operation is the concatenation. Such
descriptions correspond to the systems of recurrences or to
context-free grammars generating single words. The descriptive ... more >>>


TR23-166 | 5th November 2023
Dmytro Gavinsky

Patterned non-determinism in communication complexity

We define and study the model of patterned non-determinism in bipartite communication complexity, denoted by $PNP^{X\leftrightarrow Y}$.
It generalises the known models $UP^{X\leftrightarrow Y}$ and $FewP^{X\leftrightarrow Y}$ through relaxing the constraints on the witnessing structure of the underlying $NP^{X\leftrightarrow Y}$-protocol.
It is shown that for the case of total functions ... more >>>


TR98-066 | 3rd November 1998
Irit Dinur, Eldar Fischer, Guy Kindler, Ran Raz, Shmuel Safra

PCP Characterizations of NP: Towards a Polynomially-Small Error-Probability

This paper strengthens the low-error PCP characterization of NP, coming
closer to the ultimate BGLR conjecture. Namely, we prove that witnesses for
membership in any NP language can be verified with a constant
number of accesses, and with an error probability exponentially
small in the ... more >>>


TR21-084 | 21st June 2021
Liron Bronfman, Ron Rothblum

PCPs and Instance Compression from a Cryptographic Lens

Modern cryptography fundamentally relies on the assumption that the adversary trying to break the scheme is computationally bounded. This assumption lets us construct cryptographic protocols and primitives that are known to be impossible otherwise. In this work we explore the effect of bounding the adversary's power in other information theoretic ... more >>>


TR10-017 | 10th February 2010
Jonathan Ullman, Salil Vadhan

PCPs and the Hardness of Generating Synthetic Data

Revisions: 4

Assuming the existence of one-way functions, we show that there is no
polynomial-time, differentially private algorithm $A$ that takes a database
$D\in (\{0,1\}^d)^n$ and outputs a ``synthetic database'' $\hat{D}$ all of whose two-way
marginals are approximately equal to those of $D$. (A two-way marginal is the fraction
of database rows ... more >>>


TR13-122 | 5th September 2013
Irit Dinur, Venkatesan Guruswami

PCPs via low-degree long code and hardness for constrained hypergraph coloring

Revisions: 1

We develop new techniques to incorporate the recently proposed ``short code" (a low-degree version of the long code) into the construction and analysis of PCPs in the classical ``Label Cover + Fourier Analysis'' framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs ... more >>>


TR18-099 | 19th May 2018
Scott Aaronson

PDQP/qpoly = ALL

We show that combining two different hypothetical enhancements to quantum computation---namely, quantum advice and non-collapsing measurements---would let a quantum computer solve any decision problem whatsoever in polynomial time, even though neither enhancement yields extravagant power by itself. This complements a related result due to Raz. The proof uses locally decodable ... more >>>


TR24-042 | 22nd February 2024
Lisa Jaser, Jacobo Toran

Pebble Games and Algebraic Proof Systems Meet Again

Analyzing refutations of the well known
pebbling formulas we prove some new strong connections between pebble games and algebraic proof system, showing that
there is a parallelism between the reversible, black and black-white pebbling games on one side, and
the three algebraic proof systems NS, MC and ... more >>>


TR16-067 | 20th April 2016
Balagopal Komarath, Jayalal Sarma, Saurabh Sawlani

Pebbling Meets Coloring : Reversible Pebble Game On Trees

The reversible pebble game is a combinatorial game played on rooted DAGs. This game was introduced by Bennett (1989) motivated by applications in designing space efficient reversible algorithms. Recently, Chan (2013) showed that the reversible pebble game number of any DAG is the same as its Dymond-Tompa pebble number and ... more >>>


TR13-006 | 8th January 2013
Balagopal Komarath, Jayalal Sarma

Pebbling, Entropy and Branching Program Size Lower Bounds

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et al.(2011). Proving an exponential lower bound for the size of the non-deterministic thrifty branching programs would separate NL from P under the thrifty hypothesis. In this ... more >>>


TR15-208 | 26th December 2015
Shafi Goldwasser, Ofer Grossman

Perfect Bipartite Matching in Pseudo-Deterministic $RNC$

Revisions: 2

In this paper we present a pseudo-deterministic $RNC$ algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses $poly(n)$ processors, $poly({\log n})$ depth, $poly(\log n)$ random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That ... more >>>


TR00-080 | 24th July 2000
Marco Cesati

Perfect Code is W[1]-complete

We show that the parameterized problem Perfect Code belongs to W[1].
This result closes an old open question, because it was often
conjectured that Perfect Code could be a natural problem having
complexity degree intermediate between W[1] and W[2]. This result
also shows W[1]-membership of the parameterized problem Weighted
more >>>


TR10-201 | 21st December 2010
Samir Datta, Raghav Kulkarni, Raghunath Tewari

Perfect Matching in Bipartite Planar Graphs is in UL

Revisions: 1

We prove that Perfect Matching in bipartite planar graphs is in UL, improving upon
the previous bound of SPL (see [DKR10]) on its space complexity. We also exhibit space
complexity bounds for some related problems. Summarizing, we show that, constructing:
1. a Perfect Matching in bipartite planar graphs is in ... more >>>


TR05-097 | 31st August 2005
Jens Groth, Rafail Ostrovsky, Amit Sahai

Perfect Non-Interactive Zero Knowledge for NP

Non-interactive zero-knowledge (NIZK) systems are
fundamental cryptographic primitives used in many constructions,
including CCA2-secure cryptosystems, digital signatures, and various
cryptographic protocols. What makes them especially attractive, is
that they work equally well in a concurrent setting, which is
notoriously hard for interactive zero-knowledge protocols. However,
while for interactive zero-knowledge we ... 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 >>>


TR19-035 | 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 >>>


TR18-208 | 27th November 2018
Benny Applebaum, Prashant Nalini Vasudevan

Placing Conditional Disclosure of Secrets in the Communication Complexity Universe

Revisions: 2

In the *Conditional Disclosure of Secrets* (CDS) problem (Gertner et al., J. Comput. Syst. Sci., 2000) Alice and Bob, who hold $n$-bit inputs $x$ and $y$ respectively, wish to release a common secret $z$ to Carol (who knows both $x$ and $y$) if and only if the input $(x,y)$ satisfies ... more >>>


TR09-052 | 2nd May 2009
Fabian Wagner, Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf

Planar Graph Isomorphism is in Log-space

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and ... more >>>


TR11-009 | 21st January 2011
Samir Datta, Gautam Prakriya

Planarity Testing Revisited

Planarity Testing is the problem of determining whether a given graph is planar while planar embedding is the corresponding construction problem.
The bounded space complexity of these problems has been determined to be Logspace by Allender and Mahajan with the aid of Reingold's result . Unfortunately, the algorithm is quite ... more >>>


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

Planarity, Exclusivity, and Unambiguity

Comments: 1

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 >>>


TR11-148 | 3rd November 2011
Rohit Gurjar, Arpita Korwar, Jochen Messner, Simon Straub, Thomas Thierauf

Planarizing Gadgets for Perfect Matching do not Exist

To compare the complexity of the perfect matching problem for general graphs with that for planar graphs, one might try to come up with a reduction from the perfect matching problem to the planar perfect matching problem.
The obvious way to construct such a reduction is via a {\em planarizing ... more >>>


TR24-002 | 4th December 2023
Takashi Ishizuka

PLS is contained in PLC

Revisions: 1

Recently, Pasarkar, Papadimitriou, and Yannakakis (ITCS 2023) have introduced the new TFNP subclass called PLC that contains the class PPP; they also have proven that several search problems related to extremal combinatorial principles (e.g., Ramsey’s theorem and the Sunflower lemma) belong to PLC. This short paper shows that the class ... more >>>


TR16-151 | 26th September 2016
Amir Yehudayoff

Pointer chasing via triangular discrimination

We prove an essentially sharp $\tilde\Omega(n/k)$ lower bound on the $k$-round distributional complexity of the $k$-step pointer chasing problem under the uniform distribution, when Bob speaks first. This is an improvement over Nisan and Wigderson's $\tilde \Omega(n/k^2)$ lower bound. A key part of the proof is using triangular discrimination instead ... more >>>


TR97-008 | 16th March 1997
Noam Nisan, Ziv Bar-Yossef

Pointer Jumping Requires Concurrent Read

We consider the well known problem of determining the k'th
vertex reached by chasing pointers in a directed graph of
out-degree 1. The famous "pointer doubling" technique
provides an O(log k) parallel time algorithm on a
Concurrent-Read Exclusive-Write (CREW) PRAM. We prove that ... more >>>


TR08-090 | 6th August 2008
Ulrich Schöpp, Martin Hofmann

Pointer Programs and Undirected Reachability

Revisions: 1

We study pointer programs as a model of structured computation within LOGSPACE. Pointer programs capture the common description of LOGSPACE algorithms as programs that take as input some structured data (e.g. a graph) and that store in memory only a constant number of pointers to the input (e.g. to the ... more >>>


TR05-157 | 10th December 2005
Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo

Points on Computable Curves

The ``analyst's traveling salesman theorem'' of geometric
measure theory characterizes those subsets of Euclidean
space that are contained in curves of finite length.
This result, proven for the plane by Jones (1990) and
extended to higher-dimensional Euclidean spaces by
Okikiolu (1991), says that a bounded set $K$ is contained
more >>>


TR13-050 | 1st April 2013
Venkatesan Guruswami, Patrick Xia

Polar Codes: Speed of polarization and polynomial gap to capacity

Revisions: 1

We prove that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within $\epsilon > 0$ of the Shannon capacity with a block length, construction complexity, and decoding complexity all bounded by a *polynomial* in $1/\epsilon$. Polar coding gives the *first known explicit construction* with rigorous ... more >>>


TR12-165 | 19th November 2012
Martin Albrecht, Pooya Farshim, Jean-Charles Faugère, Gottfried Herold, Ludovic Perret

Polly Cracker, Revisited

We initiate the formal treatment of cryptographic constructions based on the hardness of computing remainders modulo an ideal in multivariate polynomial rings. Of particular interest to us is a class of schemes known as "Polly Cracker." We start by formalising and studying the relation between the ideal remainder problem and ... more >>>


TR09-011 | 31st January 2009
Mark Braverman

Poly-logarithmic independence fools AC0 circuits

We prove that poly-sized AC0 circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log^2 n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has ... more >>>


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

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

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


TR06-008 | 23rd November 2005
MohammadTaghi Hajiaghayi, Guy Kortsarz, Mohammad R. Salavatipour

Polylogarithmic Approximation Algorithm for Non-Uniform Multicommodity Buy-at-Bulk

We consider the non-uniform multicommodity buy-at-bulk network
design problem. In this problem we are given a graph $G(V,E)$ with
two cost functions on the edges, a buy cost $b:E\longrightarrow \RR^+$ and a rent cost
$r:E\longrightarrow\RR^+$, and a set of source-sink pairs $s_i,t_i\in V$ ($1\leq i\leq \alpha$)
with each pair $i$ ... more >>>


TR05-117 | 17th September 2005
Piotr Indyk, David P. Woodruff

Polylogarithmic Private Approximations and Efficient Matching

A private approximation of a function f is defined to be another function F that approximates f in the usual sense, but does not reveal any information about the input x other than what can be deduced from f(x). We give the first two-party private approximation of the Euclidean distance ... more >>>


TR04-053 | 17th June 2004
A. Pavan, N. V. Vinodchandran

Polylogarithmic Round Arthur-Merlin Games and Random-Self-Reducibility

We consider Arthur-Merlin proof systems where (a) Arthur is a probabilistic quasi-polynomial time Turing machine
and (AMQ)(b) Arthur is a probabilistic exponential time Turing machine (AME). We prove two new results related to these classes.

more >>>

TR04-007 | 13th January 2004
Alan L. Selman, Samik Sengupta

Polylogarithmic-round Interactive Proofs for coNP Collapses the Exponential Hierarchy

Revisions: 1 , Comments: 1

It is known \cite{BHZ87} that if every language in coNP has a
constant-round interactive proof system, then the polynomial hierarchy
collapses. On the other hand, Lund {\em et al}.\ \cite{LFKN92} have shown that
#SAT, the #P-complete function that outputs the number of satisfying
assignments of a Boolean ... more >>>


TR06-081 | 19th May 2006
Spyros Kontogiannis, Panagiota Panagopoulou, Paul Spirakis

Polynomial Algorithms for Approximating Nash Equilibria of Bimatrix Games

We focus on the problem of computing an $\epsilon$-Nash equilibrium of a bimatrix game, when $\epsilon$ is an absolute constant.
We present a simple algorithm for computing a $\frac{3}{4}$-Nash equilibrium for any bimatrix game in strongly polynomial time and
we next show how to extend this algorithm so as to ... more >>>


TR94-024 | 12th December 1994
Marek Karpinski, Angus Macintyre

Polynomial Bounds for VC Dimension of Sigmoidal Neural Networks

We introduce a new method for proving explicit upper bounds on the VC
Dimension of general functional basis networks, and prove as an
application, for the first time, the VC Dimension of analog neural
networks with the sigmoid activation function $\sigma(y)=1/1+e^{-y}$
to ... more >>>


TR22-043 | 2nd April 2022
Uma Girish, Kunal Mittal, Ran Raz, Wei Zhan

Polynomial Bounds On Parallel Repetition For All 3-Player Games With Binary Inputs

We prove that for every 3-player (3-prover) game $\mathcal G$ with value less than one, whose query distribution has the support $\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $\mathcal G^{\otimes n}$ decays polynomially fast to zero; that is, there ... more >>>


TR24-038 | 27th February 2024
Olaf Beyersdorff, Kaspar Kasche, Luc Nicolas Spachmann

Polynomial Calculus for Quantified Boolean Logic: Lower Bounds through Circuits and Degree

We initiate an in-depth proof-complexity analysis of polynomial calculus (Q-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of Q-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for Q-PC, similar in ... more >>>


TR24-046 | 6th March 2024
Sasank Mouli

Polynomial Calculus sizes over the Boolean and Fourier bases are incomparable

For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over $\{0,1\}$ variables of size $O(n^3polylog(n))$ but any Polynomial Calculus refutation over $\{+1,-1\}$ variables requires size $2^{\Omega(n)}$. This shows that Polynomial Calculus ... 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 >>>


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

Polynomial Data Structure Lower Bounds in the Group Model

Revisions: 1

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


TR14-018 | 13th February 2014
Arnab Bhattacharyya

Polynomial decompositions in polynomial time

Fix a prime $p$. Given a positive integer $k$, a vector of positive integers ${\bf \Delta} = (\Delta_1, \Delta_2, \dots, \Delta_k)$ and a function $\Gamma: \mathbb{F}_p^k \to \F_p$, we say that a function $P: \mathbb{F}_p^n \to \mathbb{F}_p$ is $(k,{\bf \Delta},\Gamma)$-structured if there exist polynomials $P_1, P_2, \dots, P_k:\mathbb{F}_p^n \to \mathbb{F}_p$ ... more >>>


TR03-036 | 27th April 2003
Bruce Edward Litow

Polynomial equation elimination via Tarski Algebra

The elimination
problem is classical:
implicitly express one of the variables occurring in a finite
system of polynomial equations as an algebraic function of a
designated subset of the remaining variables. Solutions to this
problem by resultants, or more comprehensively by
use of Gr\"{o}bner basis methods are available. In this ... more >>>


TR05-150 | 5th December 2005
Neeraj Kayal, Nitin Saxena

Polynomial Identity Testing for Depth 3 Circuits

We study the identity testing problem for depth $3$ arithmetic circuits ($\Sigma\Pi\Sigma$ circuits). We give the first deterministic polynomial time identity test for $\Sigma\Pi\Sigma$ circuits with bounded top fanin. We also show that the {\em rank} of a minimal and simple $\Sigma\Pi\Sigma$ circuit with bounded top fanin, computing zero, can ... more >>>


TR22-115 | 17th August 2022
Dieter van Melkebeek, Andrew Morgan

Polynomial Identity Testing via Evaluation of Rational Functions

Revisions: 1

We introduce a hitting set generator for Polynomial Identity Testing
based on evaluations of low-degree univariate rational functions at
abscissas associated with the variables. Despite the univariate
nature, we establish an equivalence up to rescaling with a generator
introduced by Shpilka and Volkovich, which has a similar structure but
uses ... more >>>


TR01-067 | 18th September 2001
Hubie Chen

Polynomial Programs and the Razborov-Smolensky Method

Representations of boolean functions as polynomials (over rings) have
been used to establish lower bounds in complexity theory. Such
representations were used to great effect by Smolensky, who
established that MOD q \notin AC^0[MOD p] (for distinct primes p, q)
by representing AC^0[MOD p] functions as low-degree multilinear
polynomials over ... more >>>


TR13-021 | 5th February 2013
Kristoffer Arnsfelt Hansen, Vladimir Podolskii

Polynomial threshold functions and Boolean threshold circuits

We study the complexity of computing Boolean functions on general
Boolean domains by polynomial threshold functions (PTFs). A typical
example of a general Boolean domain is $\{1,2\}^n$. We are mainly
interested in the length (the number of monomials) of PTFs, with
their degree and weight being of secondary interest. We ... more >>>


TR07-037 | 2nd February 2007
Leonid Gurvits

Polynomial time algorithms to approximate mixed volumes within a simply exponential factor

Revisions: 1

We study in this paper randomized algorithms to approximate the mixed volume of well-presented convex compact sets.
Our main result is a poly-time algorithm which approximates $V(K_1,...,K_n)$ with multiplicative error $e^n$ and
with better rates if the affine dimensions of most of the sets $K_i$ are small.\\
Our approach is ... more >>>


TR98-064 | 6th November 1998
Wenceslas Fernandez de la Vega, Marek Karpinski

Polynomial Time Approximation of Dense Weighted Instances of MAX-CUT

We give the first polynomial time approximability characterization
of dense weighted instances of MAX-CUT, and some other dense
weighted NP-hard problems in terms of their empirical weight
distributions. This gives also the first almost sharp
characterization of inapproximability of unweighted 0,1
MAX-BISECTION instances ... more >>>


TR01-034 | 30th April 2001
Cristina Bazgan, Wenceslas Fernandez de la Vega, Marek Karpinski

Polynomial Time Approximation Schemes for Dense Instances of Minimum Constraint Satisfaction

It is known that large fragments of the class of dense
Minimum Constraint Satisfaction (MIN-CSP) problems do not have
polynomial time approximation schemes (PTASs) contrary to their
Maximum Constraint Satisfaction analogs. In this paper we prove,
somewhat surprisingly, that the minimum satisfaction of dense
instances of kSAT-formulas, ... more >>>


TR02-025 | 26th April 2002
Wenceslas Fernandez de la Vega, Marek Karpinski, Claire Kenyon, Yuval Rabani

Polynomial Time Approximation Schemes for Metric Min-Sum Clustering

We give polynomial time approximation schemes for the problem
of partitioning an input set of n points into a fixed number
k of clusters so as to minimize the sum over all clusters of
the total pairwise distances in a cluster. Our algorithms work
for arbitrary metric spaces as well ... more >>>


TR97-024 | 9th June 1997
Marek Karpinski

Polynomial Time Approximation Schemes for Some Dense Instances of NP-Hard Optimization Problems

We survey recent results on the existence of polynomial time
approximation schemes for some dense instances of NP-hard
optimization problems. We indicate further some inherent limits
for existence of such schemes for some other dense instances of
the optimization problems.

more >>>

TR03-059 | 18th May 2003
Harumichi Nishimura, Tomoyuki Yamakami

Polynomial time quantum computation with advice

Revisions: 2

Advice is supplementary information that enhances the computational
power of an underlying computation. This paper focuses on advice that
is given in the form of a pure quantum state. The notion of advised
quantum computation has a direct connection to non-uniform quantum
circuits and tally languages. The paper examines the ... more >>>


TR94-005 | 12th December 1994
Noga Alon, Alan Frieze, Dominic Welsh

Polynomial time randomised approximation schemes for Tutte-Gr\"{o}thendieck invariants: the dense case

The Tutte-Gr\"othendieck polynomial $T(G;x,y)$ of a graph $G$
encodes numerous interesting combinatorial quantities associated
with the graph. Its evaluation in various points in the $(x,y)$
plane give the number of spanning forests of the graph, the number
of its strongly connected orientations, the number of its proper
$k$-colorings, the (all ... more >>>


TR95-039 | 11th July 1995
Tomoyuki Yamakami

Polynomial Time Samplable Distributions

Revisions: 2

This paper studies distributions which
can be ``approximated'' by sampling algorithms in time polynomial in
the length of their outputs. First, it is known that if
polynomial-time samplable distributions are polynomial-time
computable, then NP collapses to P. This paper shows by a simple
... more >>>


TR09-039 | 6th April 2009
Matei David, Periklis Papakonstantinou, Anastasios Sidiropoulos

Polynomial Time with Restricted Use of Randomness

We define a hierarchy of complexity classes that lie between P and RP, yielding a new way of quantifying partial progress towards the derandomization of RP. A standard approach in derandomization is to reduce the number of random bits an algorithm uses. We instead focus on a model of computation ... more >>>


TR10-133 | 20th August 2010
Parikshit Gopalan, Adam Klivans, Raghu Meka

Polynomial-Time Approximation Schemes for Knapsack and Related Counting Problems using Branching Programs

We give a deterministic, polynomial-time algorithm for approximately counting the number of {0,1}-solutions to any instance of the knapsack problem. On an instance of length n with total weight W and accuracy parameter eps, our algorithm produces a (1 + eps)-multiplicative approximation in time poly(n,log W,1/eps). We also give algorithms ... more >>>


TR06-082 | 18th June 2006
Prabhu Manyem

Polynomial-Time Maximisation Classes: Syntactic Hierarchy

In Descriptive Complexity, there is a vast amount of literature on
decision problems, and their classes such as \textbf{P, NP, L and NL}. ~
However, research on the descriptive complexity of optimisation problems
has been limited. In a previous paper [Man], we characterised
the optimisation versions of \textbf{P} via expressions ... more >>>


TR23-076 | 24th May 2023
Lijie Chen, Zhenjian Lu, Igor Carboni Oliveira, Hanlin Ren, Rahul Santhanam

Polynomial-Time Pseudodeterministic Construction of Primes

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time.

... more >>>

TR18-018 | 22nd January 2018
John Hitchcock, Adewale Sekoni, Hadi Shafei

Polynomial-Time Random Oracles and Separating Complexity Classes

Bennett and Gill (1981) showed that P^A != NP^A != coNP^A for a random
oracle A, with probability 1. We investigate whether this result
extends to individual polynomial-time random oracles. We consider two
notions of random oracles: p-random oracles in the sense of
martingales and resource-bounded measure (Lutz, 1992; Ambos-Spies ... more >>>



TR15-085 | 23rd May 2015
Irit Dinur, Prahladh Harsha, Guy Kindler

Polynomially Low Error PCPs with polyloglogn Queries via Modular Composition

We show that every language in NP has a PCP verifier that tosses $O(\log n)$ random coins, has perfect completeness, and a soundness error of at most $1/poly(n)$, while making at most $O(poly\log\log n)$ queries into a proof over an alphabet of size at most $n^{1/poly\log\log n}$. Previous constructions that ... more >>>


TR15-196 | 4th December 2015
Andris Ambainis

Polynomials, Quantum Query Complexity, and Grothendieck's Inequality

Revisions: 1

We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials.
Namely, a partial Boolean function $f$ is computable by a 1-query quantum algorithm with error bounded by $\epsilon<1/2$ iff $f$ can be approximated by a degree-2 polynomial with error bounded by $\epsilon'<1/2$.
This result holds for two ... more >>>


TR12-118 | 18th September 2012
Avi Wigderson, Amir Yehudayoff

Population Recovery and Partial Identification

We study several problems in which an {\em unknown} distribution over an {\em unknown} population of vectors needs to be recovered from partial or noisy samples, each of which nearly completely erases or obliterates the original vector. For example, consider a distribution $p$ over a population $V \subseteq \{0,1\}^n$. A ... more >>>


TR21-013 | 20th January 2021
Srinivasan Arunachalam, Penghui Yao

Positive spectrahedrons: Geometric properties, Invariance principles and Pseudorandom generators

In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary $m$-facet polytopes in $n$ variables with seed length poly-logarithmic in $m,n$, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, ... more >>>


TR21-038 | 15th March 2021
Alessandro Chiesa, Fermi Ma, Nicholas Spooner, Mark Zhandry

Post-Quantum Succinct Arguments

Revisions: 1

We prove that Kilian's four-message succinct argument system is post-quantum secure in the standard model when instantiated with any probabilistically checkable proof and any collapsing hash function (which in turn exist based on the post-quantum hardness of Learning with Errors).

At the heart of our proof is a new ... more >>>


TR21-167 | 23rd November 2021
Alex Lombardi, Fermi Ma, Nicholas Spooner

Post-Quantum Zero Knowledge, Revisited (or: How to Do Quantum Rewinding Undetectably)

Revisions: 1

A major difficulty in quantum rewinding is the fact that measurement is destructive: extracting information from a quantum state irreversibly changes it. This is especially problematic in the context of zero-knowledge simulation, where preserving the adversary's state is essential.

In this work, we develop new techniques for ... more >>>


TR02-028 | 15th May 2002
Eric Allender, Harry Buhrman, Michal Koucky, Detlef Ronneburger, Dieter van Melkebeek

Power from Random Strings

Revisions: 1 , Comments: 1

We consider sets of strings with high Kolmogorov complexity, mainly
in resource-bounded settings but also in the traditional
recursion-theoretic sense. We present efficient reductions, showing
that these sets are hard and complete for various complexity classes.

In particular, in addition to the usual Kolmogorov complexity measure
K, ... more >>>


TR22-186 | 31st December 2022
Prashanth Amireddy, Sai Jayasurya, Jayalal Sarma

Power of Decision Trees with Monotone Queries

In this paper, we initiate study of the computational power of adaptive and non-adaptive monotone decision trees – decision trees where each query is a monotone function on the input bits. In the most general setting, the monotone decision tree height (or size) can be viewed as a measure of ... more >>>


TR96-045 | 28th August 1996
Bernd Borchert, Lane A. Hemaspaandra, Jörg Rothe

Powers-of-Two Acceptance Suffices for Equivalence and Bounded Ambiguity Problems

Comments: 2

We study EP, the subclass of NP consisting of those
languages accepted by NP machines that when they accept always have a
number of accepting paths that is a power of two. We show that the
negation equivalence problem for OBDDs (ordered binary decision
diagrams) ... more >>>


TR02-036 | 30th May 2002
Stephen A. Fenner

PP-lowness and a simple definition of AWPP

We show that the counting classes AWPP and APP [Li 1993] are more robust
than previously thought. Our results identify asufficient condition for
a language to be low for PP, and we show that this condition is at least
as weak as other previously studied criteria. Our results imply that
more >>>


TR22-128 | 11th September 2022
Romain Bourneuf, Lukáš Folwarczný, Pavel Hubacek, Alon Rosen, Nikolaj Schwartzbach

PPP-Completeness and Extremal Combinatorics

Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey's theorem on monochromatic subgraphs and the Erdos-Rado sunflower lemma. Implicit versions of the corresponding total search problems are known to be PWPP-hard; here "implicit" means that the collection is represented ... more >>>


TR18-113 | 30th May 2018
Dominik Scheder

PPSZ for $k \geq 5$: More Is Better

We show that for $k \geq 5$, the PPSZ algorithm for $k$-SAT runs exponentially faster if there is an exponential number of satisfying assignments. More precisely, we show that for every $k\geq 5$, there is a strictly increasing function $f: [0,1] \rightarrow \mathbb{R}$ with $f(0) = 0$ that has the ... more >>>


TR21-069 | 12th May 2021
Dominik Scheder

PPSZ is better than you think

Revisions: 1

PPSZ, for long time the fastest known algorithm for k-SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to 0 or 1, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being ... more >>>


TR18-179 | 31st October 2018
Dominik Scheder

PPSZ on CSP Instances with Multiple Solutions

We study the success probability of the PPSZ algorithm on $(d,k)$-CSP formulas. We greatly simplify the analysis of Hertli, Hurbain, Millius, Moser, Szedlak, and myself for the notoriously difficult case that the input formula has more than one satisfying assignment.

more >>>

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

Pr-ZSUBEXP is not contained in Pr-RP

Revisions: 1

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


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

Practical Relativistic Zero-Knowledge for NP

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


TR17-191 | 15th December 2017
Alexander Smal, Navid Talebanfard

Prediction from Partial Information and Hindsight, an Alternative Proof

Revisions: 2

Let $X$ be a random variable distributed over $n$-bit strings with $H(X) \ge n - k$, where $k \ll n$. Using subadditivity we know that a random coordinate looks random. Meir and Wigderson [TR17-149] showed a random coordinate looks random to an adversary who is allowed to query around $n/k$ ... more >>>


TR17-149 | 7th October 2017
Or Meir, Avi Wigderson

Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds

Revisions: 5

Consider a random sequence of $n$ bits that has entropy at least $n-k$, where $k\ll n$. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random”. In this work, we prove a stronger result that ... more >>>


TR01-043 | 26th April 2001
Mikhail V. Vyugin, Vladimir Vyugin

Predictive complexity and information

A new notion of predictive complexity and corresponding amount of
information are considered.
Predictive complexity is a generalization of Kolmogorov complexity
which bounds the ability of any algorithm to predict elements of
a sequence of outcomes. We consider predictive complexity for a wide class
of bounded loss functions which ... more >>>


TR05-119 | 15th September 2005
Nadia Creignou, Phokion G. Kolaitis, Bruno Zanuttini

Preferred representations of Boolean relations

We introduce the notion of a plain basis for a co-clone in Post's lattice. Such a basis is a set of relations B such that every constraint C over a relation in the co-clone is logically equivalent to a conjunction of equalities and constraints over B and the same variables ... more >>>


TR16-172 | 3rd November 2016
William Hoza, Adam Klivans

Preserving Randomness for Adaptive Algorithms

Revisions: 4

We introduce the concept of a randomness steward, a tool for saving random bits when executing a randomized estimation algorithm $\mathrm{Est}$ on many adaptively chosen inputs. For each execution, the chosen input to $\mathrm{Est}$ remains hidden from the steward, but the steward chooses the randomness of $\mathrm{Est}$ and, crucially, is ... more >>>


TR00-047 | 29th June 2000
Tobias Polzin, Siavash Vahdati Daneshmand

Primal-Dual Approaches to the Steiner Problem

We study several old and new algorithms for computing lower
and upper bounds for the Steiner problem in networks using dual-ascent
and primal-dual strategies. These strategies have been proven to be
very useful for the algorithmic treatment of the Steiner problem. We
show that none of the known algorithms ... more >>>


TR03-071 | 18th August 2003
Markus Bläser, Andreas Jakoby, Maciej Liskiewicz, Bodo Manthey

Privacy in Non-Private Environments

Revisions: 1

We study private computations in information-theoretical settings on
networks that are not 2-connected. Non-2-connected networks are
``non-private'' in the sense that most functions cannot privately be
computed on such networks. We relax the notion of privacy by
introducing lossy private protocols, which generalize private
protocols. We measure the information each ... more >>>


TR05-141 | 29th November 2005
Amos Beimel, Paz Carmi, Kobbi Nissim, Enav Weinreb

Private Approximation of Search Problems

Many approximation algorithms have been presented in the last decades
for hard search problems. The focus of this paper is on cryptographic
applications, where it is desired to design algorithms which do not
leak unnecessary information. Specifically, we are interested in
private approximation algorithms -- efficient algorithms ... more >>>


TR03-009 | 3rd February 2003
Markus Bläser, Andreas Jakoby, Maciej Liskiewicz, Bodo Manthey

Private Computation --- $k$-connected versus $1$-connected Networks

Revisions: 1

We study the role of connectivity of communication networks in private
computations under information theoretic settings. It will be shown that
some functions can be computed by private protocols even if the
underlying network is 1-connected but not 2-connected. Then we give a
complete characterisation of non-degenerate functions that can ... more >>>


TR01-051 | 4th May 2001
Sophie Laplante, Richard Lassaigne, Frederic Magniez, Sylvain Peyronnet, Michel de Rougemont

Probabilistic abstraction for model checking: An approach based on property testing

Revisions: 1

In model checking, program correctness on all inputs is verified
by considering the transition system underlying a given program.
In practice, the system can be intractably large.
In property testing, a property of a single input is verified
by looking at a small subset of that input.
We ... more >>>


TR18-123 | 3rd July 2018
Alessandro Chiesa, Peter Manohar, Igor Shinkar

Probabilistic Checking against Non-Signaling Strategies from Linearity Testing

Revisions: 1

Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is ... more >>>


TR17-070 | 15th April 2017
Shachar Lovett, Sankeerth Rao Karingula, Alex Vardy

Probabilistic Existence of Large Sets of Designs

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t-(n,k,?)$ combinatorial design with tiny, yet ... more >>>


TR11-144 | 2nd November 2011
Greg Kuperberg, Shachar Lovett, Ron Peled

Probabilistic existence of rigid combinatorial structures

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. ... more >>>


TR22-072 | 15th May 2022
Halley Goldberg, Valentine Kabanets, Zhenjian Lu, Igor Oliveira

Probabilistic Kolmogorov Complexity with Applications to Average-Case Complexity

Understanding the relationship between the worst-case and average-case complexities of $\mathrm{NP}$ and of other subclasses of $\mathrm{PH}$ is a long-standing problem in complexity theory. Over the last few years, much progress has been achieved in this front through the investigation of meta-complexity: the complexity of problems that refer to the ... more >>>


TR17-036 | 22nd February 2017
Dean Doron, Francois Le Gall, Amnon Ta-Shma

Probabilistic logarithmic-space algorithms for Laplacian solvers

A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system $L x=b$, where $L$ is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem.
more >>>


TR00-085 | 19th September 2000
Rustam Mubarakzjanov

Probabilistic OBDDs: on Bound of Width versus Bound of Error

Ordered binary decision diagrams (OBDDs) are well established tools to
represent Boolean functions. There are a lot of results concerning
different types of generalizations of OBDDs. The same time, the power
of the most general form of OBDD, namely probabilistic (without bounded
error) OBDDs, is not studied enough. In ... more >>>


TR94-008 | 12th December 1994
Oded Goldreich

Probabilistic Proof Systems (A Survey)

Various types of probabilistic proof systems have played
a central role in the development of computer science in the last decade.
In this exposition, we concentrate on three such proof systems ---
interactive proofs, zero-knowledge proofs,
and probabilistic checkable proofs --- stressing the essential
role of randomness in each ... more >>>


TR11-136 | 12th October 2011
eran gat , shafi goldwasser

Probabilistic Search Algorithms with Unique Answers and Their Cryptographic Applications

Revisions: 1

In this paper we introduce a new type of probabilistic search algorithm, which we call the
{\it Bellagio} algorithm: a probabilistic algorithm which is guaranteed to run in expected polynomial time,
and to produce a correct and {\it unique} solution with high probability.
We argue the applicability of such algorithms ... more >>>


TR96-035 | 27th March 1996
Ronald V. Book, Heribert Vollmer, Klaus W. Wagner

Probabilistic Type-2 Operators and ``Almost''-Classes


We define and examine several probabilistic operators ranging over sets
(i.e., operators of type 2), among them the formerly studied
ALMOST-operator. We compare their power and prove that they all coincide
for a wide variety of classes. As a consequence, we characterize the
ALMOST-operator which ranges over infinite objects ... more >>>


TR01-027 | 23rd March 2001
Marius Zimand

Probabilistically Checkable Proofs The Easy Way

We present a weaker variant of the PCP Theorem that admits a
significantly easier proof. In this
variant the prover only has $n^t$ time to compute each
bit of his answer, for an arbitray but fixed constant
$t$, in contrast to
being all powerful. We show that
3SAT ... more >>>


TR98-040 | 24th July 1998
Madhu Sudan, Luca Trevisan

Probabilistically checkable proofs with low amortized query complexity

The error probability of Probabilistically Checkable Proof (PCP)
systems can be made exponentially small in the number of queries
by using sequential repetition. In this paper we are interested
in determining the precise rate at which the error goes down in
an optimal protocol, and we make substantial progress toward ... more >>>


TR24-023 | 21st January 2024
Shuichi Hirahara, Naoto Ohsaka

Probabilistically Checkable Reconfiguration Proofs and Inapproximability of Reconfiguration Problems

Motivated by the inapproximability of reconfiguration problems, we present a new PCP-type characterization of PSPACE, which we call a probabilistically checkable reconfiguration proof (PCRP): Any PSPACE computation can be encoded into an exponentially long sequence of polynomially long proofs such that every adjacent pair of the proofs differs in at ... more >>>


TR01-063 | 5th August 2001
Michal Parnas, Dana Ron, Alex Samorodnitsky

Proclaiming Dictators and Juntas or Testing Boolean Formulae

Revisions: 1

We consider the problem of determining whether a given
function f : {0,1}^n -> {0,1} belongs to a certain class
of Boolean functions F or whether it is far from the class.
More precisely, given query access to the function f and given
a distance parameter epsilon, we would ... more >>>


TR23-133 | 13th September 2023
David Ellis, Guy Kindler, Noam Lifshitz, Dor Minzer

Product mixing in compact Lie groups

Revisions: 2

If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is ... more >>>


TR09-101 | 20th October 2009
Nitin Saxena

Progress on Polynomial Identity Testing

Polynomial identity testing (PIT) is the problem of checking whether a given
arithmetic circuit is the zero circuit. PIT ranks as one of the most important
open problems in the intersection of algebra and computational complexity. In the last
few years, there has been an impressive progress on this ... more >>>


TR13-186 | 27th December 2013
Nitin Saxena

Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.

more >>>

TR16-183 | 16th November 2016
Joshua Brakensiek, Venkatesan Guruswami

Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy

Revisions: 2

A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or NP-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $\Gamma$ consists of a pair $(\Psi_P, ... more >>>


TR04-098 | 5th November 2004
Lance Fortnow, Rahul Santhanam, Luca Trevisan

Promise Hierarchies

We show that for any constant a, ZPP/b(n) strictly contains
ZPTIME(n^a)/b(n) for some b(n) = O(log n log log n). Our techniques
are very general and give the same hierarchy for all the common
promise time classes including RTIME, NTIME \cap coNTIME, UTIME,
MATIME, AMTIME and BQTIME.

We show a ... more >>>


TR21-043 | 15th March 2021
Peter Dixon, A. Pavan, N. V. Vinodchandran

Promise Problems Meet Pseudodeterminism

The Acceptance Probability Estimation Problem (APEP) is to additively approximate the acceptance probability of a Boolean circuit. This problem admits a probabilistic approximation scheme. A central question is whether we can design a pseudodeterministic approximation algorithm for this problem: a probabilistic polynomial-time algorithm that outputs a canonical approximation with high ... more >>>


TR16-098 | 16th June 2016
Michael Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson

Proof Complexity Lower Bounds from Algebraic Circuit Complexity


We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the ... more >>>


TR20-184 | 10th December 2020
Dmitry Itsykson, Artur Riazanov

Proof complexity of natural formulas via communication arguments

A canonical communication problem ${\rm Search}(\phi)$ is defined for every unsatisfiable CNF $\phi$: an assignment to the variables of $\phi$ is distributed among the communicating parties, they are to find a clause of $\phi$ falsified by this assignment. Lower bounds on the randomized $k$-party communication complexity of ${\rm Search}(\phi)$ in ... more >>>


TR24-030 | 22nd February 2024
Olaf Beyersdorff, Tim Hoffmann, Luc Nicolas Spachmann

Proof Complexity of Propositional Model Counting

Recently, the proof system MICE for the model counting problem #SAT was introduced by Fichte, Hecher and Roland (SAT’22). As demonstrated by Fichte et al., the system MICE can be used for proof logging for state-of-the-art #SAT solvers.
We perform a proof-complexity study of MICE. For this we first simplify ... more >>>


TR14-120 | 16th September 2014
Olaf Beyersdorff, Leroy Chew, Mikolas Janota

Proof Complexity of Resolution-based QBF Calculi

Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of important
QBF solvers. However, the proof complexity of these proof systems is currently not well understood and in particular
lower bound techniques are missing. In this paper we exhibit a new and elegant proof technique ... 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 >>>


TR23-053 | 19th April 2023
Leroy Chew

Proof Simulation via Round-based Strategy Extraction for QBF

The proof complexity of Quantified Boolean Formulas (QBF) relates to both QBF solving and OBF certification. One method to p-simulate a QBF proof system is by formalising the soundness of its strategy extraction in propositional logic. In this work we illustrate how to use extended QBF Frege to simulate LD-Q(Drrs)-Res, ... more >>>


TR09-092 | 8th October 2009
Olaf Beyersdorff, Johannes Köbler, Sebastian Müller

Proof Systems that Take Advice

One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow (JSL 79), where they defined
propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajicek (JSL 07) have ... more >>>


TR98-008 | 15th January 1998
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, Mario Szegedy

Proof verification and the hardness of approximation problems.


We show that every language in NP has a probablistic verifier
that checks membership proofs for it using
logarithmic number of random bits and by examining a
<em> constant </em> number of bits in the proof.
If a string is in the language, then there exists a proof ... more >>>


TR12-120 | 24th September 2012
Boaz Barak

Proof vs. Truth in Computational Complexity

Revisions: 1

In this survey, I discuss the general question of what evidence can we use to predict the answer for open questions in computational complexity, as well as the concrete evidence currently known for two conjectures: Khot's Unique Games Conjecture and Feige's Random 3SAT Hypothesis.

more >>>

TR15-024 | 16th February 2015
Oded Goldreich, Tom Gur, Ron Rothblum

Proofs of Proximity for Context-Free Languages and Read-Once Branching Programs

Proofs of proximity are probabilistic proof systems in which the verifier only queries a sub-linear number of input bits, and soundness only means that, with high probability, the input is close to an accepting input. In their minimal form, called Merlin-Arthur proofs of proximity (MAP), the verifier receives, in addition ... more >>>


TR17-155 | 13th October 2017
Alessandro Chiesa, Tom Gur

Proofs of Proximity for Distribution Testing

Revisions: 1

Distribution testing is an area of property testing that studies algorithms that receive few samples from a probability distribution D and decide whether D has a certain property or is far (in total variation distance) from all distributions with that property. Most natural properties of distributions, however, require a large ... more >>>


TR17-114 | 1st July 2017
Maciej Li\'skiewicz, Matthias Lutter, Rüdiger Reischuk

Proper Learning of k-term DNF Formulas from Satisfying Assignments

In certain applications there may only be positive samples available to
to learn concepts of a class of interest,
and this has to be done properly, i.e. the
hypothesis space has to coincide with the concept class,
and without false positives, i.e. the hypothesis always has be a subset ... more >>>


TR15-040 | 24th March 2015
Shay Moran, Amir Yehudayoff

Proper PAC learning is compressing

Revisions: 1

We prove that proper PAC learnability implies compression. Namely, if a concept $C \subseteq \Sigma^X$ is properly PAC learnable with $d$ samples, then $C$ has a sample compression scheme of size $2^{O(d)}$.
In particular, every boolean concept class with constant VC dimension has a sample compression scheme of constant size. ... more >>>


TR12-163 | 24th November 2012
Avishay Tal

Properties and Applications of Boolean Function Composition

For Boolean functions $f:\{0,1\}^n \to \{0,1\}$ and $g:\{0,1\}^m \to \{0,1\}$, the function composition of $f$ and $g$ denoted by $f\circ g : \{0,1\}^{nm} \to \{0,1\}$ is the value of $f$ on $n$ inputs, each of them is the calculation of $g$ on a distinct set of $m$ Boolean variables. Motivated ... more >>>


TR04-019 | 15th January 2004
Christian Glaßer, A. Pavan, Alan L. Selman, Samik Sengupta

Properties of NP-Complete Sets

We study several properties of sets that are complete for NP.
We prove that if $L$ is an NP-complete set and $S \not\supseteq L$ is a p-selective sparse set, then $L - S$ is many-one-hard for NP. We demonstrate existence of a sparse set $S \in \mathrm{DTIME}(2^{2^{n}})$
such ... more >>>


TR04-096 | 4th November 2004
Eldar Fischer, Frederic Magniez, Michel de Rougemont

Property and Equivalence Testing on Strings

Revisions: 1

Using a new statistical embedding of words which has similarities with the Parikh mapping, we first construct a tolerant tester for the equality of two words, whose complexity is independent of the string size, where the distance between inputs is measured by the normalized edit distance with moves. As a ... more >>>


TR96-057 | 18th November 1996
Oded Goldreich, Dana Ron

Property Testing and its connection to Learning and Approximation


In this paper, we consider the question of determining whether
a function $f$ has property $P$ or is $\e$-far from any
function with property $P$.
The property testing algorithm is given a sample of the value
of $f$ on instances drawn according to some distribution.
In some cases,
more >>>


TR13-142 | 11th October 2013
Abhishek Bhrushundi, Sourav Chakraborty, Raghav Kulkarni

Property Testing Bounds for Linear and Quadratic Functions via Parity Decision Trees

Revisions: 2

In this paper, we study linear and quadratic Boolean functions in the context of property testing. We do this by observing that the query complexity of testing properties of linear and quadratic functions can be characterized in terms of the complexity in another model of computation called parity decision trees. ... more >>>


TR11-045 | 1st April 2011
Eric Blais, Joshua Brody, Kevin Matulef

Property Testing Lower Bounds via Communication Complexity

Revisions: 1

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in ... more >>>


TR08-040 | 3rd April 2008
Sourav Chakraborty, Laszlo Babai

Property Testing of Equivalence under a Permutation Group Action

For a permutation group $G$ acting on the set $\Omega$
we say that two strings $x,y\,:\,\Omega\to\boole$
are {\em $G$-isomorphic} if they are equivalent under
the action of $G$, \ie, if for some $\pi\in G$ we have
$x(i^{\pi})=y(i)$ for all $i\in\Omega$.
Cyclic Shift, Graph Isomorphism ... more >>>


TR13-187 | 27th December 2013
Peyman Afshani, Kevin Matulef, Bryan Wilkinson

Property Testing on Linked Lists

Revisions: 1

We define a new property testing model for algorithms that do not have arbitrary query access to the input, but must instead traverse it in a manner that respects the underlying data structure in which it is stored. In particular, we consider the case when the underlying data structure is ... more >>>


TR14-042 | 2nd April 2014
Kashyap Dixit, Deeparnab Chakrabarty, Madhav Jha, C. Seshadhri

Property Testing on Product Distributions: Optimal Testers for Bounded Derivative Properties

The primary problem in property testing is to decide whether a given function satisfies a certain property, or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the {\em uniform} distribution on the domain. ... more >>>


TR10-156 | 24th October 2010
Victor Chen, Madhu Sudan, Ning Xie

Property Testing via Set-Theoretic Operations

Given two testable properties $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, under what conditions are the union, intersection or set-difference
of these two properties also testable?
We initiate a systematic study of these basic set-theoretic operations in the context of property
testing. As an application, we give a conceptually different proof that linearity is ... more >>>


TR16-007 | 23rd January 2016
Guy Kindler

Property Testing, PCP, andJuntas

Revisions: 1

The first part of this thesis strengthens the low-error PCP
characterization of NP, coming closer to the upper limit of the
conjecture of~\cite{BGLR}.

In the second part we show that a boolean function over
$n$ variables can be tested for the property of depending ... more >>>


TR98-067 | 12th November 1998
Paul Beame

Propositional Proof Complexity: Past, Present and Future

Proof complexity, the study of the lengths of proofs in
propositional logic, is an area of study that is fundamentally connected
both to major open questions of computational complexity theory and
to practical properties of automated theorem provers. In the last
decade, there have been a number of significant advances ... more >>>


TR23-066 | 4th May 2023
Klim Efremenko, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena

Protecting Single-Hop Radio Networks from Message Drops

Single-hop radio networks (SHRN) are a well studied abstraction of communication over a wireless channel. In this model, in every round, each of the $n$ participating parties may decide to broadcast a message to all the others, potentially causing collisions. We consider the SHRN model in the presence of stochastic ... more >>>


TR23-142 | 21st September 2023
Tom Gur, Wilfred Salmon, Sergii Strelchuk

Provable Advantage in Quantum PAC Learning

We revisit the problem of characterising the complexity of Quantum PAC learning, as introduced by Bshouty and Jackson [SIAM J. Comput.
1998, 28, 1136–1153]. Several quantum advantages have been demonstrated in this setting, however, none are generic: they apply to particular concept classes and typically only work when the distribution ... more >>>


TR21-180 | 21st December 2021
Noga Ron-Zewi, Ron Rothblum

Proving as Fast as Computing: Succinct Arguments with Constant Prover Overhead

Succinct arguments are proof systems that allow a powerful, but untrusted, prover to convince a weak verifier that an input $x$ belongs to a language $L \in NP$, with communication that is much shorter than the $NP$ witness. Such arguments, which grew out of the theory literature, are now drawing ... more >>>


TR05-080 | 21st July 2005
Michael R. Fellows, Frances A. Rosamond, Udi Rotics, Stefan Szeider

Proving NP-hardness for clique-width I: non-approximability of sequential clique-width

Revisions: 1

Clique-width is a graph parameter, defined by a composition mechanism
for vertex-labeled graphs, which measures in a certain sense the
complexity of a graph. Hard graph problems (e.g., problems
expressible in Monadic Second Order Logic, that includes NP-hard
problems) can be solved efficiently for graphs of certified small
clique-width. It ... more >>>


TR05-081 | 21st July 2005
Michael R. Fellows, Frances A. Rosamond, Udi Rotics, Stefan Szeider

Proving NP-hardness for clique-width II: non-approximability of clique-width

Revisions: 1

Clique-width is a graph parameter that measures in a certain sense the
complexity of a graph. Hard graph problems (e.g., problems
expressible in Monadic Second Order Logic with second-order
quantification on vertex sets, that includes NP-hard problems) can be
solved efficiently for graphs of certified small clique-width. It is
widely ... more >>>


TR18-003 | 2nd January 2018
Roei Tell

Proving that prBPP=prP is as hard as "almost" proving that P \ne NP

Revisions: 5

We show that any proof that $promise\textrm{-}\mathcal{BPP}=promise\textrm{-}\mathcal{P}$ necessitates proving circuit lower bounds that almost yield that $\mathcal{P}\ne\mathcal{NP}$. More accurately, we show that if $promise\textrm{-}\mathcal{BPP}=promise\textrm{-}\mathcal{P}$, then for essentially any super-constant function $f(n)=\omega(1)$ it holds that $NTIME[n^{f(n)}]\not\subseteq\mathcal{P}/\mathrm{poly}$. The conclusion of the foregoing conditional statement cannot be improved (to conclude that $\mathcal{NP}\not\subseteq\mathcal{P}/\mathrm{poly}$) without ... more >>>


TR23-016 | 22nd February 2023
Yuval Filmus, Edward Hirsch, Artur Riazanov, Alexander Smal, Marc Vinyals

Proving Unsatisfiability with Hitting Formulas

Hitting formulas have been studied in many different contexts at least since [Iwama 1989]. A hitting formula is a set of Boolean clauses such that any two of the clauses cannot be simultaneously falsified. [Peitl and Szeider 2022] conjectured that the family of unsatisfiable hitting formulas should contain the hardest ... more >>>


TR20-083 | 30th May 2020
Eli Ben-Sasson, Dan Carmon, Yuval Ishai, Swastik Kopparty, Shubhangi Saraf

Proximity Gaps for Reed-Solomon Codes

Revisions: 3

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are $\delta$-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are $\delta$-close to the property. In particular, no set ... more >>>


TR10-058 | 7th April 2010
Oded Goldreich, Tali Kaufman

Proximity Oblivious Testing and the Role of Invariances

Revisions: 1

We present a general notion of properties that
are characterized by local conditions that are invariant
under a sufficiently rich class of symmetries.
Our framework generalizes two popular models of
testing graph properties as well as the algebraic
invariances studied by Kaufman and Sudan (STOC'08).

We show that, in the ... more >>>


TR18-122 | 3rd July 2018
Igor Carboni Oliveira, Rahul Santhanam

Pseudo-derandomizing learning and approximation

We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser
[GG11]. A pseudo-deterministic algorithm is a probabilistic algorithm which produces a fixed
output with high probability. We explore pseudo-determinism in the settings of learning and ap-
proximation. Our goal is to simulate known randomized algorithms in these settings ... more >>>


TR17-105 | 14th June 2017
Shafi Goldwasser, Ofer Grossman, Dhiraj Holden

Pseudo-Deterministic Proofs

We introduce pseudo-deterministic interactive proofs (psdAM): interactive proof systems for search problems where
the verifier is guaranteed with high probability to output the same output on different executions.
As in the case with classical interactive proofs,
the verifier is a probabilistic polynomial time algorithm interacting with an untrusted powerful prover.

... more >>>

TR19-177 | 6th December 2019
Shafi Goldwasser, Ofer Grossman, Sidhanth Mohanty, David Woodruff

Pseudo-deterministic Streaming

A pseudo-deterministic algorithm is a (randomized) algorithm which, when run multiple times on the same input, with high probability outputs the same result on all executions. Classic streaming algorithms, such as those for finding heavy hitters, approximate counting, $\ell_2$ approximation, finding a nonzero entry in a vector (for turnstile algorithms) ... 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 >>>


TR12-119 | 24th September 2012
Ilario Bonacina, Nicola Galesi

Pseudo-partitions, Transversality and Locality: A Combinatorial Characterization for the Space Measure in Algebraic Proof Systems

We devise a new combinatorial characterization for proving space lower bounds in algebraic systems like Polynomial Calculus (Pc) and Polynomial Calculus with Resolution (Pcr). Our method can be thought as a Spoiler-Duplicator game, which is capturing boolean reasoning on polynomials instead that clauses as in the case of Resolution. Hence, ... more >>>


TR01-064 | 10th September 2001
Moni Naor, Omer Reingold, Alon Rosen

Pseudo-Random Functions and Factoring

Factoring integers is the most established problem on which
cryptographic primitives are based. This work presents an efficient
construction of {\em pseudorandom functions} whose security is based
on the intractability of factoring. In particular, we are able to
construct efficient length-preserving pseudorandom functions where
each evaluation requires only a ... more >>>


TR21-132 | 11th September 2021
Eric Binnendyk

Pseudo-random functions and uniform learnability

Revisions: 1

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


TR20-151 | 8th October 2020
Venkatesan Guruswami, Vinayak Kumar

Pseudobinomiality of the Sticky Random Walk

Random walks on expanders are a central and versatile tool in pseudorandomness. If an arbitrary half of the vertices of an expander graph are marked, known Chernoff bounds for expander walks imply that the number $M$ of marked vertices visited in a long $n$-step random walk strongly concentrates around the ... more >>>


TR21-039 | 15th March 2021
Zhenjian Lu, Igor Carboni Oliveira, Rahul Santhanam

Pseudodeterministic Algorithms and the Structure of Probabilistic Time

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $BPTIME$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows.

1. A new pseudorandom generator and its consequences: We build on techniques developed to ... more >>>


TR16-196 | 5th December 2016
Igor Carboni Oliveira, Rahul Santhanam

Pseudodeterministic Constructions in Subexponential Time

We study {\it pseudodeterministic constructions}, i.e., randomized algorithms which output the {\it same solution} on most computation paths. We establish unconditionally that there is an infinite sequence $\{p_n\}_{n \in \mathbb{N}}$ of increasing primes and a randomized algorithm $A$ running in expected sub-exponential time such that for each $n$, on input ... more >>>


TR21-019 | 17th February 2021
Edward Pyne, Salil Vadhan

Pseudodistributions That Beat All Pseudorandom Generators

Revisions: 1

A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ... more >>>


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 >>>


TR05-043 | 5th April 2005
Emanuele Viola

Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates

We exhibit an explicitly computable `pseudorandom' generator stretching $l$ bits into $m(l) = l^{\Omega(\log l)}$ bits that look random to constant-depth circuits of size $m(l)$ with $\log m(l)$ arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS '93) that achieves the same ... more >>>


TR17-122 | 28th July 2017
Rohit Gurjar, Ben Lee Volk

Pseudorandom Bits for Oblivious Branching Programs

We construct a pseudorandom generator which fools read-$k$ oblivious branching programs and, more generally, any linear length oblivious branching program, assuming that the sequence according to which the bits are read is known in advance. For polynomial width branching programs, the seed lengths in our constructions are $\tilde{O}(n^{1-1/2^{k-1}})$ (for the ... more >>>


TR07-081 | 10th August 2007
Andrej Bogdanov, Emanuele Viola

Pseudorandom bits for polynomials

We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators $G : \F^s \to \F^n$ that fool polynomials over a prime field $\F$:
\begin{enumerate}
\item a ... more >>>


TR18-028 | 7th February 2018
Xin Li

Pseudorandom Correlation Breakers, Independence Preserving Mergers and their Applications

Revisions: 1

The recent line of study on randomness extractors has been a great success, resulting in exciting new techniques, new connections, and breakthroughs to long standing open problems in the following five seemingly different topics: seeded non-malleable extractors, privacy amplification protocols with an active adversary, independent source extractors (and explicit Ramsey ... more >>>


TR17-113 | 1st July 2017
Andrej Bogdanov, Alon Rosen

Pseudorandom Functions: Three Decades Later

In 1984, Goldreich, Goldwasser and Micali formalized the concept of pseudorandom functions and proposed a construction based on any length-doubling pseudorandom generator. Since then, pseudorandom functions have turned out to be an extremely influential abstraction, with applications ranging from message authentication to barriers in proving computational complexity lower bounds.

In ... more >>>


TR10-033 | 6th March 2010
Shachar Lovett, Partha Mukhopadhyay, Amir Shpilka

Pseudorandom generators for $\mathrm{CC}_0[p]$ and the Fourier spectrum of low-degree polynomials over finite fields

In this paper we give the first construction of a pseudorandom generator, with seed length $O(\log n)$, for $\mathrm{CC}_0[p]$, the class of constant-depth circuits with unbounded fan-in $\mathrm{MOD}_p$ gates, for some prime $p$. More accurately, the seed length of our generator is $O(\log n)$ for any constant error $\epsilon>0$. In ... more >>>


TR10-168 | 9th November 2010
Thomas Watson

Pseudorandom Generators for Combinatorial Checkerboards

Revisions: 2

We define a combinatorial checkerboard to be a function $f:\{1,\ldots,m\}^d\to\{1,-1\}$ of the form $f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)$ for some functions $f_i:\{1,\ldots,m\}\to\{1,-1\}$. This is a variant of combinatorial rectangles, which can be defined in the same way but using $\{0,1\}$ instead of $\{1,-1\}$. We consider the problem of constructing explicit pseudorandom generators for combinatorial ... more >>>


TR10-176 | 15th November 2010
Parikshit Gopalan, Raghu Meka, Omer Reingold, David Zuckerman

Pseudorandom Generators for Combinatorial Shapes

Revisions: 1

We construct pseudorandom generators for combinatorial shapes, which substantially generalize combinatorial rectangles, small-bias spaces, 0/1 halfspaces, and 0/1 modular sums. A function $f:[m]^n \rightarrow \{0,1\}^n$ is an $(m,n)$-combinatorial shape if there exist sets $A_1,\ldots,A_n \subseteq [m]$ and a symmetric function $h:\{0,1\}^n \rightarrow \{0,1\}$ such that $f(x_1,\ldots,x_n) = h(1_{A_1} (x_1),\ldots,1_{A_n}(x_n))$. Our ... more >>>


TR10-113 | 16th July 2010
Michal Koucky, Prajakta Nimbhorkar, Pavel Pudlak

Pseudorandom Generators for Group Products

We prove that the pseudorandom generator introduced in Impagliazzo et al. (1994) fools group products of a given finite group. The seed length is $O(\log n \log 1 / \epsilon)$, where $n$ is the length of the word and $\epsilon$ is the error. The result is equivalent to the statement ... more >>>


TR13-155 | 10th November 2013
Gil Cohen, Amnon Ta-Shma

Pseudorandom Generators for Low Degree Polynomials from Algebraic Geometry Codes

Revisions: 2

Constructing pseudorandom generators for low degree polynomials has received a considerable attention in the past decade. Viola [CC 2009], following an exciting line of research, constructed a pseudorandom generator for degree d polynomials in n variables, over any prime field. The seed length used is $O(d \log{n} + d 2^d)$, ... more >>>


TR17-025 | 16th February 2017
Pooya Hatami, Avishay Tal

Pseudorandom Generators for Low-Sensitivity Functions

A Boolean function is said to have maximal sensitivity $s$ if $s$ is the largest number of Hamming neighbors of a point which differ from it in function value. We construct a pseudorandom generator with seed-length $2^{O(\sqrt{s})} \cdot \log(n)$ that fools Boolean functions on $n$ variables with maximal sensitivity at ... more >>>


TR18-147 | 19th August 2018
Michael Forbes, Zander Kelley

Pseudorandom Generators for Read-Once Branching Programs, in any Order

A central question in derandomization is whether randomized logspace (RL) equals deterministic logspace (L). To show that RL=L, it suffices to construct explicit pseudorandom generators (PRGs) that fool polynomial-size read-once (oblivious) branching programs (roBPs). Starting with the work of Nisan, pseudorandom generators with seed-length $O(\log^2 n)$ were constructed. Unfortunately, ... more >>>


TR10-035 | 7th March 2010
Mark Braverman, Anup Rao, Ran Raz, Amir Yehudayoff

Pseudorandom Generators for Regular Branching Programs

We give new pseudorandom generators for \emph{regular} read-once branching programs of small width.
A branching program is regular if the in-degree of every vertex in it is (0 or) $2$.
For every width $d$ and length $n$,
our pseudorandom generator uses a seed of length $O((\log d + \log\log n ... more >>>


TR20-138 | 9th September 2020
William Hoza, Edward Pyne, Salil Vadhan

Pseudorandom Generators for Unbounded-Width Permutation Branching Programs

Revisions: 1

We prove that the Impagliazzo-Nisan-Wigderson (STOC 1994) pseudorandom generator (PRG) fools ordered (read-once) permutation branching programs of unbounded width with a seed length of $\widetilde{O}(\log d + \log n \cdot \log(1/\varepsilon))$, assuming the program has only one accepting vertex in the final layer. Here, $n$ is the length of the ... more >>>


TR18-112 | 5th June 2018
Raghu Meka, Omer Reingold, Avishay Tal

Pseudorandom Generators for Width-3 Branching Programs

Revisions: 1

We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/\epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work ... more >>>


TR18-015 | 25th January 2018
Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett

Pseudorandom Generators from Polarizing Random Walks

Revisions: 1 , Comments: 1

We propose a new framework for constructing pseudorandom generators for $n$-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in $[-1,1]^n$. Next, we use a fractional pseudorandom generator as steps of a random walk in $[-1,1]^n$ that ... more >>>


TR18-155 | 8th September 2018
Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, Avishay Tal

Pseudorandom generators from the second Fourier level and applications to AC0 with parity gates

A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design ... more >>>


TR00-023 | 11th May 2000
Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

Pseudorandom Generators in Propositional Proof Complexity

We call a pseudorandom generator $G_n:\{0,1\}^n\to \{0,1\}^m$ {\em
hard} for a propositional proof system $P$ if $P$ can not efficiently
prove the (properly encoded) statement $G_n(x_1,\ldots,x_n)\neq b$ for
{\em any} string $b\in\{0,1\}^m$. We consider a variety of
``combinatorial'' pseudorandom generators inspired by the
Nisan-Wigderson generator on the one hand, and ... more >>>


TR11-007 | 17th January 2011
Benny Applebaum

Pseudorandom Generators with Long Stretch and Low locality from Random Local One-Way Functions

Revisions: 3

We continue the study of pseudorandom generators (PRG) $G:\{0,1\}^n \rightarrow \{0,1\}^m$ in NC0. While it is known that such generators are likely to exist for the case of small sub-linear stretch $m=n+n^{1-\epsilon}$, it remains unclear whether achieving larger stretch such as $m=2n$ or even $m=n+n^2$ is possible. The existence of ... more >>>


TR98-074 | 16th December 1998
Madhu Sudan, Luca Trevisan, Salil Vadhan

Pseudorandom generators without the XOR Lemma

Revisions: 2


Impagliazzo and Wigderson have recently shown that
if there exists a decision problem solvable in time $2^{O(n)}$
and having circuit complexity $2^{\Omega(n)}$
(for all but finitely many $n$) then $\p=\bpp$. This result
is a culmination of a series of works showing
connections between the existence of hard predicates
and ... more >>>


TR95-006 | 1st January 1995
Kenneth W. Regan, D. Sivakumar, Jin-Yi Cai

Pseudorandom Generators, Measure Theory, and Natural Proofs


This paper proves that if strong pseudorandom number generators or
one-way functions exist, then the class of languages that have
polynomial-sized circuits is not small within exponential
time, in terms of the resource-bounded measure theory of Lutz.
More precisely, if for some \epsilon > 0 there exist nonuniformly
2^{n^{\epsilon}}-hard ... 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 >>>


TR10-129 | 16th August 2010
Jeff Kinne, Dieter van Melkebeek, Ronen Shaltiel

Pseudorandom Generators, Typically-Correct Derandomization, and Circuit Lower Bounds

The area of derandomization attempts to provide efficient deterministic simulations of randomized algorithms in various algorithmic settings. Goldreich and Wigderson introduced a notion of "typically-correct" deterministic simulations, which are allowed to err on few inputs. In this paper we further the study of typically-correct derandomization in two ways.

First, we ... more >>>


TR21-170 | 25th November 2021
Reyad Abed Elrazik, Robert Robere, Assaf Schuster, Gal Yehuda

Pseudorandom Self-Reductions for NP-Complete Problems

A language $L$ is random-self-reducible if deciding membership in $L$ can be reduced (in polynomial time) to deciding membership in $L$ for uniformly random instances. It is known that several "number theoretic" languages (such as computing the permanent of a matrix) admit random self-reductions. Feigenbaum and Fortnow showed that NP-complete ... more >>>


TR18-006 | 10th January 2018
Subhash Khot, Dor Minzer, Muli Safra

Pseudorandom Sets in Grassmann Graph have Near-Perfect Expansion

Revisions: 2

We prove that pseudorandom sets in Grassmann graph have near-perfect expansion as hypothesized in [DKKMS-2]. This completes
the proof of the $2$-to-$2$ Games Conjecture (albeit with imperfect completeness) as proposed in [KMS, DKKMS-1], along with a
contribution from [BKT].

The Grassmann graph $Gr_{global}$ contains induced subgraphs $Gr_{local}$ that are themselves ... more >>>


TR05-022 | 19th February 2005
Omer Reingold, Luca Trevisan, Salil Vadhan

Pseudorandom Walks in Biregular Graphs and the RL vs. L Problem

Motivated by Reingold's recent deterministic log-space algorithm for Undirected S-T Connectivity (ECCC TR 04-94), we revisit the general RL vs. L question, obtaining the following results.

1. We exhibit a new complete problem for RL: S-T Connectivity restricted to directed graphs for which the random walk is promised to have ... more >>>


TR06-013 | 24th January 2006
Luca Trevisan

Pseudorandomness and Combinatorial Constructions

In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or unknown. In computer science,
probabilistic algorithms are sometimes simpler and more efficient
than the best known ... more >>>


TR14-076 | 27th May 2014
Thomas Steinke

Pseudorandomness and Fourier Growth Bounds for Width 3 Branching Programs

Revisions: 1

We present an explicit pseudorandom generator for oblivious, read-once, width-$3$ branching programs, which can read their input bits in any order. The generator has seed length $\tilde{O}( \log^3 n ).$ The previously best known seed length for this model is $n^{1/2+o(1)}$ due to Impagliazzo, Meka, and Zuckerman (FOCS '12). Our ... more >>>


TR19-155 | 6th November 2019
Rahul Santhanam

Pseudorandomness and the Minimum Circuit Size Problem

We explore the possibility of basing one-way functions on the average-case hardness of the fundamental Minimum Circuit Size Problem (MCSP[$s$]), which asks whether a Boolean function on $n$ bits specified by its truth table has circuits of size $s(n)$.

1. (Pseudorandomness from Zero-Error Average-Case Hardness) We show that for ... more >>>


TR04-086 | 12th October 2004
Ronen Shaltiel, Chris Umans

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. Examples are Arthur-Merlin games and approximate counting and sampling of NP-witnesses. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.

Our main technical contribution allows one to ``boost'' a given hardness assumption. One special ... more >>>


TR13-132 | 23rd September 2013
Michael Forbes, Ramprasad Saptharishi, Amir Shpilka

Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the ... more >>>


TR12-083 | 29th June 2012
Thomas Steinke

Pseudorandomness for Permutation Branching Programs Without the Group Theory

We exhibit an explicit pseudorandom generator that stretches an $O \left( \left( w^4 \log w + \log (1/\varepsilon) \right) \cdot \log n \right)$-bit random seed to $n$ pseudorandom bits that cannot be distinguished from truly random bits by a permutation branching program of width $w$ with probability more than $\varepsilon$. ... more >>>


TR11-117 | 3rd September 2011
Andrej Bogdanov, Periklis Papakonstantinou, Andrew Wan

Pseudorandomness for read-once formulas

We give an explicit construction of a pseudorandom generator for read-once formulas whose inputs can be read in arbitrary order. For formulas in $n$ inputs and arbitrary gates of fan-in at most $d = O(n/\log n)$, the pseudorandom generator uses $(1 - \Omega(1))n$ bits of randomness and produces an output ... more >>>


TR13-086 | 13th June 2013
Omer Reingold, Thomas Steinke, Salil Vadhan

Pseudorandomness for Regular Branching Programs via Fourier Analysis

Revisions: 1

We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is $O(\log^2 n)$, where $n$ is the length of the branching program. The previous best seed length known for this model was $n^{1/2+o(1)}$, ... more >>>


TR09-070 | 1st September 2009
Andrej Bogdanov, Zeev Dvir, Elad Verbin, Amir Yehudayoff

Pseudorandomness for Width 2 Branching Programs

Bogdanov and Viola (FOCS 2007) constructed a pseudorandom
generator that fools degree $k$ polynomials over $\F_2$ for an arbitrary
constant $k$. We show that such generators can also be used to fool branching programs of width 2 and polynomial length that read $k$ bits of inputs at a
time. This ... more >>>


TR12-057 | 7th May 2012
Russell Impagliazzo, Raghu Meka, David Zuckerman

Pseudorandomness from Shrinkage

Revisions: 2

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models ... more >>>


TR22-024 | 17th February 2022
Louis Golowich, Salil Vadhan

Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs

Revisions: 1

We study the pseudorandomness of random walks on expander graphs against tests computed by symmetric functions and permutation branching programs. These questions are motivated by applications of expander walks in the coding theory and derandomization literatures. We show that expander walks fool symmetric functions up to a $O(\lambda)$ error in ... more >>>


TR16-037 | 15th March 2016
Sergei Artemenko, Russell Impagliazzo, Valentine Kabanets, Ronen Shaltiel

Pseudorandomness when the odds are against you

Impagliazzo and Wigderson showed that if $\text{E}=\text{DTIME}(2^{O(n)})$ requires size $2^{\Omega(n)}$ circuits, then
every time $T$ constant-error randomized algorithm can be simulated deterministically in time $\poly(T)$. However, such polynomial slowdown is a deal breaker when $T=2^{\alpha \cdot n}$, for a constant $\alpha>0$, as is the case for some randomized algorithms for ... more >>>


TR03-028 | 28th February 2003
Olivier Powell

PSPACE contains almost complete problems

An almost complete set A for a complexity class C is a language of C which is not complete, but that has the property that ``many'' languages of C reduce to A, where the term ``many'' is used in reference to Lutz's resource bounded measure (rbm). The question of the ... more >>>


TR07-021 | 5th March 2007
Brett Hemenway, Rafail Ostrovsky

Public Key Encryption Which is Simultaneously a Locally-Decodable Error-Correcting Code

Revisions: 3

In this paper, we introduce the notion of a Public-Key Encryption (PKE) Scheme that is also a Locally-Decodable Error-Correcting Code.
In particular, our construction simultaneously satisfies all of the following properties:
\begin{itemize}
\item
Our Public-Key Encryption is semantically secure under a certain number-theoretic hardness assumption
... more >>>


TR11-118 | 6th September 2011
Brett Hemenway, Rafail Ostrovsky, Martin Strauss, Mary Wootters

Public Key Locally Decodable Codes with Short Keys

This work considers locally decodable codes in the computationally bounded channel model. The computationally bounded channel model, introduced by Lipton in 1994, views the channel as an adversary which is restricted to polynomial-time computation. Assuming the existence of IND-CPA secure public-key encryption, we present a construction of public-key locally decodable ... more >>>


TR13-130 | 17th September 2013
Mark Braverman, Ankit Garg

Public vs private coin in bounded-round information

Revisions: 1

We precisely characterize the role of private randomness in the ability of Alice to send a message to Bob while minimizing the amount of information revealed to him. We show that if using private randomness a message can be transmitted while revealing $I$ bits of information, the transmission can be ... more >>>


TR21-029 | 1st March 2021
Inbar Kaslasi, Ron Rothblum, Prashant Nalini Vasudevan

Public-Coin Statistical Zero-Knowledge Batch Verification against Malicious Verifiers

Suppose that a problem $\Pi$ has a statistical zero-knowledge (SZK) proof with communication complexity $m$. The question of batch verification for SZK asks whether one can prove that $k$ instances $x_1,\ldots,x_k$ all belong to $\Pi$ with a statistical zero-knowledge proof whose communication complexity is better than $k \cdot m$ (which ... more >>>


TR02-042 | 7th June 2002
Dima Grigoriev

Public-key cryptography and invariant theory

Public-key crypto
Contact: dima@maths.univ-rennes1.fr
Author: Dima Grigoriev
Title: Public-key cryptography and invariant theory
Abstract: Public-key cryptosystems are suggested based on invariants of group
representations

more >>>

TR96-056 | 12th November 1996
Oded Goldreich, Shai Halevi

Public-Key Cryptosystems from Lattice Reduction Problems

We present a new proposal for a trapdoor one-way function,
from which we derive public-key encryption and digital
signatures. The security of the new construction is based
on the conjectured computational difficulty of lattice-reduction
problems, providing a possible alternative to existing
more >>>


TR08-100 | 14th November 2008
Chris Peikert

Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem

We construct public-key cryptosystems that are secure assuming the
\emph{worst-case} hardness of approximating the length of a shortest
nonzero vector in an $n$-dimensional lattice to within a small
$\poly(n)$ factor. Prior cryptosystems with worst-case connections
were based either on the shortest vector problem for a \emph{special
class} of lattices ... more >>>


TR22-011 | 25th January 2022
Andrej Bogdanov, Miguel Cueto Noval, Charlotte Hoffmann, Alon Rosen

Public-Key Encryption from Continuous LWE

The continuous learning with errors (CLWE) problem was recently introduced by Bruna
et al. (STOC 2021). They showed that its hardness implies infeasibility of learning Gaussian
mixture models, while its tractability implies efficient Discrete Gaussian Sampling and thus
asymptotic improvements in worst-case lattice algorithms. No reduction between CLWE and
LWE ... more >>>


TR23-098 | 5th July 2023
Andrej Bogdanov, Pravesh Kothari, Alon Rosen

Public-Key Encryption, Local Pseudorandom Generators, and the Low-Degree Method

The low-degree method postulates that no efficient algorithm outperforms low-degree polynomials in certain hypothesis-testing tasks. It has been used to understand computational indistinguishability in high-dimensional statistics.

We explore the use of the low-degree method in the context of cryptography. To this end, we apply it in the design and analysis ... more >>>


TR12-130 | 3rd October 2012
Abuzer Yakaryilmaz

Public-qubits versus private-coins

We introduce a new public quantum interactive proof system, namely qAM, by augmenting the verifier with a fixed-size quantum register in Arthur-Merlin game. We focus on space-bounded verifiers, and compare our new public system with private-coin interactive proof (IP) system in the same space bounds. We show that qAM systems ... more >>>


TR21-139 | 24th September 2021
Venkatesan Guruswami, Jonathan Mosheiff

Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity

Revisions: 2

We prove the existence of Reed-Solomon codes of any desired rate $R \in (0,1)$ that are combinatorially list-decodable up to a radius approaching $1-R$, which is the information-theoretic limit. This is established by starting with the full-length $[q,k]_q$ Reed-Solomon code over a field $\mathbb{F}_q$ that is polynomially larger than the ... more >>>


TR05-031 | 1st March 2005
Carme Alvarez, Joaquim Gabarro, Maria Serna

Pure Nash equilibria in games with a large number of actions

We study the computational complexity of deciding the existence of a
Pure Nash Equilibrium in multi-player strategic games.
We address two fundamental questions: how can we represent a game?, and
how can we represent a game with polynomial pay-off functions?
Our results show that the computational complexity of
deciding ... more >>>


TR01-081 | 8th August 2001
Palash Sarkar

Pushdown Automaton with the Ability to Flip its Stack

We introduce the idea of pushdown automata with the ability to flip
its stack. By bounding the number of times the stack can be flipped
we obtain a hierarchy of language classes from the context free
languages to the recursively enumerable languages. We show that each
class in ... more >>>




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