196 research outputs found
On the Implicit Graph Conjecture
The implicit graph conjecture states that every sufficiently small,
hereditary graph class has a labeling scheme with a polynomial-time computable
label decoder. We approach this conjecture by investigating classes of label
decoders defined in terms of complexity classes such as P and EXP. For
instance, GP denotes the class of graph classes that have a labeling scheme
with a polynomial-time computable label decoder. Until now it was not even
known whether GP is a strict subset of GR. We show that this is indeed the case
and reveal a strict hierarchy akin to classical complexity. We also show that
classes such as GP can be characterized in terms of graph parameters. This
could mean that certain algorithmic problems are feasible on every graph class
in GP. Lastly, we define a more restrictive class of label decoders using
first-order logic that already contains many natural graph classes such as
forests and interval graphs. We give an alternative characterization of this
class in terms of directed acyclic graphs. By showing that some small,
hereditary graph class cannot be expressed with such label decoders a weaker
form of the implicit graph conjecture could be disproven.Comment: 13 pages, MFCS 201
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Extending and Applying a Framework for the Cryptographic Verification of Java Programs
Abstract. In our previous work, we have proposed a framework which allows tools that can check standard noninterference properties but a priori cannot deal with cryptography to establish cryptographic indistinguishability properties, such as privacy properties, for Java programs. We refer to this framework as the CVJ framework (Cryptographic Verification of Java Programs) in this paper. While so far the CVJ framework directly supports public-key encryption (without corruption and without a public-key infrastructure) only, in this work we further instantiate the framework to support, among others, public-key encryption and digital signatures, both with corruption and a public-key infrastructure, as well as (private) symmetric encryption. Since these cryptographic primitives are very common in security-critical applications, our extensions make the framework much more widely applicable. To illustrate the usefulness and applicability of the extensions proposed in this paper, we apply the framework along with the tool Joana, which allows for the fully automatic verification of noninterference properties of Java programs, to establish cryptographic privacy properties of a (non-trivial) cloud storage application, where clients can store private information on a remote server.
Spectral tensor-train decomposition
The accurate approximation of high-dimensional functions is an essential task
in uncertainty quantification and many other fields. We propose a new function
approximation scheme based on a spectral extension of the tensor-train (TT)
decomposition. We first define a functional version of the TT decomposition and
analyze its properties. We obtain results on the convergence of the
decomposition, revealing links between the regularity of the function, the
dimension of the input space, and the TT ranks. We also show that the
regularity of the target function is preserved by the univariate functions
(i.e., the "cores") comprising the functional TT decomposition. This result
motivates an approximation scheme employing polynomial approximations of the
cores. For functions with appropriate regularity, the resulting
\textit{spectral tensor-train decomposition} combines the favorable
dimension-scaling of the TT decomposition with the spectral convergence rate of
polynomial approximations, yielding efficient and accurate surrogates for
high-dimensional functions. To construct these decompositions, we use the
sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of
tensors resulting from suitable discretizations of the target function. We
assess the performance of the method on a range of numerical examples: a
modifed set of Genz functions with dimension up to , and functions with
mixed Fourier modes or with local features. We observe significant improvements
in performance over an anisotropic adaptive Smolyak approach. The method is
also used to approximate the solution of an elliptic PDE with random input
data. The open source software and examples presented in this work are
available online.Comment: 33 pages, 19 figure
Applications of Derandomization Theory in Coding
Randomized techniques play a fundamental role in theoretical computer science
and discrete mathematics, in particular for the design of efficient algorithms
and construction of combinatorial objects. The basic goal in derandomization
theory is to eliminate or reduce the need for randomness in such randomized
constructions. In this thesis, we explore some applications of the fundamental
notions in derandomization theory to problems outside the core of theoretical
computer science, and in particular, certain problems related to coding theory.
First, we consider the wiretap channel problem which involves a communication
system in which an intruder can eavesdrop a limited portion of the
transmissions, and construct efficient and information-theoretically optimal
communication protocols for this model. Then we consider the combinatorial
group testing problem. In this classical problem, one aims to determine a set
of defective items within a large population by asking a number of queries,
where each query reveals whether a defective item is present within a specified
group of items. We use randomness condensers to explicitly construct optimal,
or nearly optimal, group testing schemes for a setting where the query outcomes
can be highly unreliable, as well as the threshold model where a query returns
positive if the number of defectives pass a certain threshold. Finally, we
design ensembles of error-correcting codes that achieve the
information-theoretic capacity of a large class of communication channels, and
then use the obtained ensembles for construction of explicit capacity achieving
codes.
[This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi
A universal median quasi-Monte Carlo integration
We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit
cube in several weighted function spaces with different smoothness classes. We
consider approximating the integral of a function by the median of several
integral estimates under independent and random choices of the underlying QMC
point sets (either linearly scrambled digital nets or infinite-precision
polynomial lattice point sets). Even though our approach does not require any
information on the smoothness and weights of a target function space as an
input, we can prove a probabilistic upper bound on the worst-case error for the
respective weighted function space, where the failure probability converges to
0 exponentially fast as the number of estimates increases. Our obtained rates
of convergence are nearly optimal for function spaces with finite smoothness,
and we can attain a dimension-independent super-polynomial convergence for a
class of infinitely differentiable functions. This implies that our
median-based QMC rule is universal in the sense that it does not need to be
adjusted to the smoothness and the weights of the function spaces and yet
exhibits the nearly optimal rate of convergence. Numerical experiments support
our theoretical results.Comment: Major revision, 32 pages, 4 figure
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