267,592 research outputs found
Parameterized complexity of machine scheduling: 15 open problems
Machine scheduling problems are a long-time key domain of algorithms and
complexity research. A novel approach to machine scheduling problems are
fixed-parameter algorithms. To stimulate this thriving research direction, we
propose 15 open questions in this area whose resolution we expect to lead to
the discovery of new approaches and techniques both in scheduling and
parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
Quantum algorithms for problems in number theory, algebraic geometry, and group theory
Quantum computers can execute algorithms that sometimes dramatically
outperform classical computation. Undoubtedly the best-known example of this is
Shor's discovery of an efficient quantum algorithm for factoring integers,
whereas the same problem appears to be intractable on classical computers.
Understanding what other computational problems can be solved significantly
faster using quantum algorithms is one of the major challenges in the theory of
quantum computation, and such algorithms motivate the formidable task of
building a large-scale quantum computer. This article will review the current
state of quantum algorithms, focusing on algorithms for problems with an
algebraic flavor that achieve an apparent superpolynomial speedup over
classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in
Quantum Computation/Information at Kinki Universit
Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields
A modified Green operator is proposed as an improvement of Fourier-based
numerical schemes commonly used for computing the electrical or thermal
response of heterogeneous media. Contrary to other methods, the number of
iterations necessary to achieve convergence tends to a finite value when the
contrast of properties between the phases becomes infinite. Furthermore, it is
shown that the method produces much more accurate local fields inside
highly-conducting and quasi-insulating phases, as well as in the vicinity of
the phases interfaces. These good properties stem from the discretization of
Green's function, which is consistent with the pixel grid while retaining the
local nature of the operator that acts on the polarization field. Finally, a
fast implementation of the "direct scheme" of Moulinec et al. (1994) that
allows for parcimonious memory use is proposed.Comment: v2: `postprint' document (a few remaining typos in the published
version herein corrected in red; results unchanged
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