REPORTS > 2017:
All reports in year 2017:
TR17-034 | 21st February 2017
Karl Bringmann, Christian Ikenmeyer, Jeroen Zuiddam

On algebraic branching programs of small width

In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the ... more >>>

TR17-033 | 19th February 2017
Daniel Kane, Shachar Lovett, Sankeerth Rao

Labeling the complete bipartite graph with no zero cycles

Assume that the edges of the complete bipartite graph $K_{n,n}$ are labeled with elements of $\mathbb{F}_2^d$, such that the sum over
any simple cycle is nonzero. What is the smallest possible value of $d$? This problem was raised by Gopalan et al. [SODA 2017] as it characterizes the alphabet size ... more >>>

TR17-032 | 17th February 2017
Olaf Beyersdorff, Joshua Blinkhorn

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

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

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

TR17-031 | 15th February 2017
Thomas Watson

Quadratic Simulations of Merlin-Arthur Games

The known proofs of $\text{MA}\subseteq\text{PP}$ incur a quadratic overhead in the running time. We prove that this quadratic overhead is necessary for black-box simulations; in particular, we obtain an oracle relative to which $\text{MA-TIME}(t)\not\subseteq\text{P-TIME}(o(t^2))$. We also show that 2-sided-error Merlin--Arthur games can be simulated by 1-sided-error Arthur--Merlin games with quadratic ... more >>>

TR17-030 | 15th February 2017
Amey Bhangale, Subhash Khot, Devanathan Thiruvenkatachari

An Improved Dictatorship Test with Perfect Completeness

A Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is called a dictator if it depends on exactly one variable i.e $f(x_1, x_2, \ldots, x_n) = x_i$ for some $i\in [n]$. In this work, we study a $k$-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems.

... more >>>

TR17-029 | 18th February 2017
Clement Canonne, Tom Gur

An Adaptivity Hierarchy Theorem for Property Testing

Revisions: 1

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of \emph{adaptive} testing algorithms, wherein each query may be determined by the answers received to prior queries, and their \emph{non-adaptive} counterparts, in which all ... more >>>

TR17-028 | 17th February 2017
Mrinal Kumar

A quadratic lower bound for homogeneous algebraic branching programs

Revisions: 1

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

TR17-027 | 16th February 2017
Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, Amnon Ta-Shma

A reduction from efficient non-malleable extractors to low-error two-source extractors with arbitrary constant rate

We show a reduction from the existence of explicit t-non-malleable
extractors with a small seed length, to the construction of explicit
two-source extractors with small error for sources with arbitrarily
small constant rate. Previously, such a reduction was known either
when one source had entropy rate above half [Raz05] or ... more >>>

TR17-026 | 17th February 2017
Valentine Kabanets, Daniel Kane, Zhenjian Lu

A Polynomial Restriction Lemma with Applications

A polynomial threshold function (PTF) of degree $d$ is a boolean function of the form $f=\mathrm{sgn}(p)$, where $p$ is a degree-$d$ polynomial, and $\mathrm{sgn}$ is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is ... 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 >>>

TR17-024 | 16th February 2017
Mika Göös, Pritish Kamath, Toniann Pitassi, Thomas Watson

Query-to-Communication Lifting for P^NP

We prove that the $\text{P}^{\small\text{NP}}$-type query complexity (alternatively, decision list width) of any boolean function $f$ is quadratically related to the $\text{P}^{\small\text{NP}}$-type communication complexity of a lifted version of $f$. As an application, we show that a certain "product" lower bound method of Impagliazzo and Williams (CCC 2010) fails to ... more >>>

TR17-023 | 15th February 2017
Russell Impagliazzo, Valentine Kabanets, Ilya Volkovich

The Power of Natural Properties as Oracles

We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP).
We obtain new circuit lower bounds, as well as some hardness results for the relativized version ... more >>>

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

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

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

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

Reconstruction of full rank Algebraic Branching Programs

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

TR17-020 | 12th February 2017
Ran Raz

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

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

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

TR17-019 | 8th February 2017
Andreas Krebs, Nutan Limaye, Michael Ludwig

A Unified Method for Placing Problems in Polylogarithmic Depth

Revisions: 1

In this work we consider the term evaluation problem which involves, given a term over some algebra and a valid input to the term, computing the value of the term on that input. This is a classical problem studied under many names such as formula evaluation problem, formula value problem ... more >>>

TR17-018 | 6th February 2017
Oded Goldreich, Guy Rothblum

Simple doubly-efficient interactive proof systems for locally-characterizable sets

A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier's strategy can be implemented in almost-linear-time.

We present direct constructions of doubly-efficient interactive proof systems for problems in $\cal P$ that are believed to have relatively high complexity.
Specifically, such ... more >>>

TR17-017 | 5th February 2017
Michal Moshkovitz, Dana Moshkovitz

Mixing Implies Lower Bounds for Space Bounded Learning

One can learn any hypothesis class $H$ with $O(\log|H|)$ labeled examples. Alas, learning with so few examples requires saving the examples in memory, and this requires $|X|^{O(\log|H|)}$ memory states, where $X$ is the set of all labeled examples. A question that arises is how many labeled examples are needed in ... more >>>

TR17-016 | 31st January 2017
Vishwas Bhargava, Gábor Ivanyos, Rajat Mittal, Nitin Saxena

Irreducibility and deterministic r-th root finding over finite fields

Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\F_q$ of characteristic $p$ (equivalently, constructing the bigger field $\F_{q^{r^e}}$). Both these problems have famous randomized ... more >>>

TR17-015 | 4th February 2017
Ilan Komargodski, Moni Naor, Eylon Yogev

White-Box vs. Black-Box Complexity of Search Problems: Ramsey and Graph Property Testing

Ramsey theory assures us that in any graph there is a clique or independent set of a certain size, roughly logarithmic in the graph size. But how difficult is it to find the clique or independent set? If the graph is given explicitly, then it is possible to do so ... more >>>

TR17-014 | 23rd January 2017
Arkadev Chattopadhyay, Michal Koucky, Bruno Loff, Sagnik Mukhopadhyay

Composition and Simulation Theorems via Pseudo-random Properties

We prove a randomized communication-complexity lower bound for a composed OrderedSearch $\circ$ IP — by lifting the randomized query-complexity lower-bound of OrderedSearch to the communication-complexity setting. We do this by extending ideas from a paper of Raz and Wigderson. We think that the techniques we develop will be useful in ... more >>>

TR17-013 | 23rd January 2017
Abhishek Bhrushundi, Prahladh Harsha, Srikanth Srinivasan

On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function F$:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be F$_k:\{0,1\}^n \rightarrow \{0,2^{k-1}\}$ by $F_k(x) = 2^{k-F(x)}$ (mod $2^k$). We consider the fractional agreement (which we refer to as $\gamma_{d,k}(F)$) of $F_k$ with ... more >>>

TR17-012 | 17th January 2017
Dominik Barth, Moritz Beck, Titus Dose, Christian Glaßer, Larissa Michler, Marc Technau

Emptiness Problems for Integer Circuits

We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, ... more >>>

TR17-011 | 22nd January 2017
Boaz Barak, Pravesh Kothari, David Steurer

Quantum entanglement, sum of squares, and the log rank conjecture

For every constant $\epsilon>0$, we give an $\exp(\tilde{O}(\sqrt{n}))$-time algorithm for the $1$ vs $1-\epsilon$ Best Separable State (BSS) problem of distinguishing, given an $n^2\times n^2$ matrix $M$ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state $\rho$ that $M$ accepts with probability $1$, ... more >>>

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

Revisions: 1

In this note we show that the Raz-McKenzie simulation algorithm which lifts deterministic query lower bounds to deterministic communication lower bounds can be implemented for functions $f$ composed with the Inner Product gadget $g_{IP}(x,y) = \sum_i x_iy_i \mathrm{mod} \, 2$ of logarithmic size. In other words, given a function $f: ... more >>> TR17-009 | 19th January 2017 Joshua Grochow, Mrinal Kumar, Michael Saks, Shubhangi Saraf Towards an algebraic natural proofs barrier via polynomial identity testing We observe that a certain kind of algebraic proof - which covers essentially all known algebraic circuit lower bounds to date - cannot be used to prove lower bounds against VP if and only if what we call succinct hitting sets exist for VP. This is analogous to the Razborov-Rudich ... more >>> TR17-008 | 14th January 2017 Benny Applebaum, Naama Haramaty, Yuval Ishai, Eyal Kushilevitz, Vinod Vaikuntanathan Low-Complexity Cryptographic Hash Functions Cryptographic hash functions are efficiently computable functions that shrink a long input into a shorter output while achieving some of the useful security properties of a random function. The most common type of such hash functions is {\em collision resistant} hash functions (CRH), which prevent an efficient attacker from finding ... more >>> TR17-007 | 19th January 2017 Michael Forbes, Amir Shpilka, Ben Lee Volk Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been ... more >>> TR17-006 | 15th December 2016 Constantinos Daskalakis, Nishanth Dikkala, Gautam Kamath Testing Ising Models Given samples from an unknown multivariate distribution$p$, is it possible to distinguish whether$p$is the product of its marginals versus$p$being far from every product distribution? Similarly, is it possible to distinguish whether$p$equals a given distribution$q$versus$p$and$q$being far from each ... more >>> TR17-005 | 10th January 2017 Nir Bitansky Verifiable Random Functions from Non-Interactive Witness-Indistinguishable Proofs Revisions: 3 Verifiable random functions (VRFs) are pseudorandom functions where the owner of the seed, in addition to computing the function's value$y$at any point$x$, can also generate a non-interactive proof$\pi$that$y$is correct (relative to so), without compromising pseudorandomness at other points. Being a natural primitive with ... more >>> TR17-004 | 8th January 2017 Scott Aaronson P=?NP In 1955, John Nash sent a remarkable letter to the National Security Agency, in which—seeking to build theoretical foundations for cryptography—he all but formulated what today we call the P=?NP problem, considered one of the great open problems of science. Here I survey the status of this problem in 2017, ... more >>> TR17-003 | 24th November 2016 Yi Deng Magic Adversaries Versus Individual Reduction: Science Wins Either Way Revisions: 1 We prove that \emph{at least} one of the following statements is true: -- (Infinitely-often) Public-key encryption and key agreement can be based on injective one-way functions; -- For every inverse polynomial$\epsilon$, the 4-round protocol from [Feige and Shamir, STOC 90] is distributional concurrent zero knowledge for any ... more >>> TR17-002 | 6th January 2017 Frantisek Duris Some notes on two lower bound methods for communication complexity We compare two methods for proving lower bounds on standard two-party model of communication complexity, the Rank method and Fooling set method. We present bounds on the number of functions$f(x,y)$,$x,y\in\{0,1\}^n$, with rank of size$k$and fooling set of size at least k,$k\in [1,2^n]\$. Using these bounds ... more >>>

TR17-001 | 6th January 2017
Stephen Cook, Bruce Kapron

A Survey of Classes of Primitive Recursive Functions

This paper is a transcription of mimeographed course notes titled A Survey of Classes of Primitive Recursive Functions", by S.A. Cook, for the University of California Berkeley course Math 290, Sect. 14, January 1967. The notes present a survey of subrecursive function
classes (and classes of relations based on these ... more >>>

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