76,770 research outputs found
Parameter Compilation
In resolving instances of a computational problem, if multiple instances of
interest share a feature in common, it may be fruitful to compile this feature
into a format that allows for more efficient resolution, even if the
compilation is relatively expensive. In this article, we introduce a formal
framework for classifying problems according to their compilability. The basic
object in our framework is that of a parameterized problem, which here is a
language along with a parameterization---a map which provides, for each
instance, a so-called parameter on which compilation may be performed. Our
framework is positioned within the paradigm of parameterized complexity, and
our notions are relatable to established concepts in the theory of
parameterized complexity. Indeed, we view our framework as playing a unifying
role, integrating together parameterized complexity and compilability theory
A Preliminary Investigation of Satisfiability Problems Not Harder than 1-in-3-SAT
The parameterized satisfiability problem over a set of Boolean
relations Gamma (SAT(Gamma)) is the problem of determining
whether a conjunctive formula over Gamma has at least one
model. Due to Schaefer\u27s dichotomy theorem the computational
complexity of SAT(Gamma), modulo polynomial-time reductions, has
been completely determined: SAT(Gamma) is always either tractable
or NP-complete. More recently, the problem of studying the
relationship between the complexity of the NP-complete cases of
SAT(Gamma) with restricted notions of reductions has attracted
attention. For example, Impagliazzo et al. studied the complexity of
k-SAT and proved that the worst-case time complexity increases
infinitely often for larger values of k, unless 3-SAT is solvable in
subexponential time. In a similar line of research Jonsson et al.
studied the complexity of SAT(Gamma) with algebraic tools borrowed
from clone theory and proved that there exists an NP-complete problem
SAT(R^{neq,neq,neq,01}_{1/3}) such that there cannot exist any NP-complete SAT(Gamma) problem with strictly lower worst-case time complexity: the easiest NP-complete SAT(Gamma) problem. In this paper
we are interested in classifying the NP-complete SAT(Gamma)
problems whose worst-case time complexity is lower than 1-in-3-SAT but
higher than the easiest problem SAT(R^{neq,neq,neq,01}_{1/3}). Recently it was conjectured that there only exists three satisfiability problems of this form. We prove that this conjecture does not hold and that there is an infinite number of such SAT(Gamma) problems. In the process we determine several algebraic properties of 1-in-3-SAT and related problems, which could be of independent interest for constructing exponential-time algorithms
A Unified Subspace Classification Framework Developed for Diagnostic System Using Microwave Signal
Subspace learning is widely used in many signal processing and statistical learning problems where the signal is assumably generated from a low dimensional space. In this paper, we present a unified classifier including several concepts from different subspace techniques, such as PCA, LRC, LDA, GLRT, etc. The objective is to project the original signal (usually of high dimension) into a smaller subspace with 1) within-class data structure preserved and 2) between-class-distance enhanced. A novel classification technique called Maximum Angle Subspace Classifier (MASC) is presented to achieve these purposes. To compensate for the computational complexity and non-convexity of MASC, an approximation is proposed as a trade-off between the classification performance and the computational issue. The approaches are applied to the problem of classifying high dimensional frequency measurements from a microwave based diagnostic system and results are compared with existing methods
The foundations of spectral computations via the Solvability Complexity Index hierarchy: Part I
The problem of computing spectra of operators is arguably one of the most
investigated areas of computational mathematics. Recent progress and the
current paper reveal that, unlike the finite-dimensional case,
infinite-dimensional problems yield a highly intricate infinite classification
theory determining which spectral problems can be solved and with which type of
algorithms. Classifying spectral problems and providing optimal algorithms is
uncharted territory in the foundations of computational mathematics. This paper
is the first of a two-part series establishing the foundations of computational
spectral theory through the Solvability Complexity Index (SCI) hierarchy and
has three purposes. First, we establish answers to many longstanding open
questions on the existence of algorithms. We show that for large classes of
partial differential operators on unbounded domains, spectra can be computed
with error control from point sampling operator coefficients. Further results
include computing spectra of operators on graphs with error control, the
spectral gap problem, spectral classifications, and discrete spectra,
multiplicities and eigenspaces. Second, these classifications determine which
types of problems can be used in computer-assisted proofs. The theory for this
is virtually non-existent, and we provide some of the first results in this
infinite classification theory. Third, our proofs are constructive, yielding a
library of new algorithms and techniques that handle problems that before were
out of reach. We show several examples on contemporary problems in the physical
sciences. Our approach is closely related to Smale's program on the foundations
of computational mathematics initiated in the 1980s, as many spectral problems
can only be computed via several limits, a phenomenon shared with the
foundations of polynomial root finding with rational maps, as proved by
McMullen
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