128 research outputs found

### INTERDEPENDENCES BETWEEN THE PUBLIC ADMINISTRATION AND PRIVATE BUSINESS ENVIRONMENT IN THE CONTEXT OF ECONOMIC CRISIS

The main aim of the UE is to protect and sustain its citizen’s expectations by harmonizing the national politics and strategies in order to generate better results. At this moment we have three main actors: Private Business Environment (PBE), National Pubbusiness, public administration, strategy, integrated system, economic crisis

### Deciding Orthogonality in Construction-A Lattices

Lattices are discrete mathematical objects with widespread applications to
integer programs as well as modern cryptography. A fundamental problem in both
domains is the Closest Vector Problem (popularly known as CVP). It is
well-known that CVP can be easily solved in lattices that have an orthogonal
basis \emph{if} the orthogonal basis is specified. This motivates the
orthogonality decision problem: verify whether a given lattice has an
orthogonal basis. Surprisingly, the orthogonality decision problem is not known
to be either NP-complete or in P.
In this paper, we focus on the orthogonality decision problem for a
well-known family of lattices, namely Construction-A lattices. These are
lattices of the form $C+q\mathbb{Z}^n$, where $C$ is an error-correcting
$q$-ary code, and are studied in communication settings. We provide a complete
characterization of lattices obtained from binary and ternary codes using
Construction-A that have an orthogonal basis. We use this characterization to
give an efficient algorithm to solve the orthogonality decision problem. Our
algorithm also finds an orthogonal basis if one exists for this family of
lattices. We believe that these results could provide a better understanding of
the complexity of the orthogonality decision problem for general lattices

### Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation

We construct efficient data structures that are resilient against a constant
fraction of adversarial noise. Our model requires that the decoder answers most
queries correctly with high probability and for the remaining queries, the
decoder with high probability either answers correctly or declares "don't
know." Furthermore, if there is no noise on the data structure, it answers all
queries correctly with high probability. Our model is the common generalization
of a model proposed recently by de Wolf and the notion of "relaxed locally
decodable codes" developed in the PCP literature.
We measure the efficiency of a data structure in terms of its length,
measured by the number of bits in its representation, and query-answering time,
measured by the number of bit-probes to the (possibly corrupted)
representation. In this work, we study two data structure problems: membership
and polynomial evaluation. We show that these two problems have constructions
that are simultaneously efficient and error-correcting.Comment: An abridged version of this paper appears in STACS 201

### Nearly Optimal Sparse Group Testing

Group testing is the process of pooling arbitrary subsets from a set of $n$
items so as to identify, with a minimal number of tests, a "small" subset of
$d$ defective items. In "classical" non-adaptive group testing, it is known
that when $d$ is substantially smaller than $n$, $\Theta(d\log(n))$ tests are
both information-theoretically necessary and sufficient to guarantee recovery
with high probability. Group testing schemes in the literature meeting this
bound require most items to be tested $\Omega(\log(n))$ times, and most tests
to incorporate $\Omega(n/d)$ items.
Motivated by physical considerations, we study group testing models in which
the testing procedure is constrained to be "sparse". Specifically, we consider
(separately) scenarios in which (a) items are finitely divisible and hence may
participate in at most $\gamma \in o(\log(n))$ tests; or (b) tests are
size-constrained to pool no more than $\rho \in o(n/d)$items per test. For both
scenarios we provide information-theoretic lower bounds on the number of tests
required to guarantee high probability recovery. In both scenarios we provide
both randomized constructions (under both $\epsilon$-error and zero-error
reconstruction guarantees) and explicit constructions of designs with
computationally efficient reconstruction algorithms that require a number of
tests that are optimal up to constant or small polynomial factors in some
regimes of $n, d, \gamma,$ and $\rho$. The randomized design/reconstruction
algorithm in the $\rho$-sized test scenario is universal -- independent of the
value of $d$, as long as $\rho \in o(n/d)$. We also investigate the effect of
unreliability/noise in test outcomes. For the full abstract, please see the
full text PDF

### Separations of Matroid Freeness Properties

Properties of Boolean functions on the hypercube invariant with respect to
linear transformations of the domain are among the most well-studied properties
in the context of property testing. In this paper, we study the fundamental
class of linear-invariant properties called matroid freeness properties. These
properties have been conjectured to essentially coincide with all testable
linear-invariant properties, and a recent sequence of works has established
testability for increasingly larger subclasses. One question left open,
however, is whether the infinitely many syntactically different properties
recently shown testable in fact correspond to new, semantically distinct ones.
This is a crucial issue since it has also been shown that there exist
subclasses of these properties for which an infinite set of syntactically
different representations collapse into one of a small, finite set of
properties, all previously known to be testable.
An important question is therefore to understand the semantics of matroid
freeness properties, and in particular when two syntactically different
properties are truly distinct. We shed light on this problem by developing a
method for determining the relation between two matroid freeness properties P
and Q. Furthermore, we show that there is a natural subclass of matroid
freeness properties such that for any two properties P and Q from this
subclass, a strong dichotomy must hold: either P is contained in Q or the two
properties are "well separated." As an application of this method, we exhibit
new, infinite hierarchies of testable matroid freeness properties such that at
each level of the hierarchy, there are functions that are far from all
functions lying in lower levels of the hierarchy. Our key technical tool is an
apparently new notion of maps between linear matroids, called matroid
homomorphisms, that might be of independent interest

### A local decision test for sparse polynomials

An ℓ-sparse (multivariate) polynomial is a polynomial containing at most ℓ-monomials in its explicit description. We assume that a polynomial is implicitly represented as a black-box: on an input query from the domain, the black-box replies with the evaluation of the polynomial at that input. We provide an efficient, randomized algorithm, that can decide whether a polynomial [MathML] given as a black-box is ℓ-sparse or not, provided that q is large compared to the polynomial's total degree. The algorithm makes only queries, which is independent of the domain size. The running time of our algorithm (in the bit-complexity model) is , where d is an upper bound on the degree of each variable. Existing interpolation algorithms for polynomials in the same model run in time . We provide a similar test for polynomials with integer coefficients

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