6,860 research outputs found
N-fold integer programming in cubic time
N-fold integer programming is a fundamental problem with a variety of natural
applications in operations research and statistics. Moreover, it is universal
and provides a new, variable-dimension, parametrization of all of integer
programming. The fastest algorithm for -fold integer programming predating
the present article runs in time with the binary length of
the numerical part of the input and the so-called Graver complexity of
the bimatrix defining the system. In this article we provide a drastic
improvement and establish an algorithm which runs in time having
cubic dependency on regardless of the bimatrix . Our algorithm can be
extended to separable convex piecewise affine objectives as well, and also to
systems defined by bimatrices with variable entries. Moreover, it can be used
to define a hierarchy of approximations for any integer programming problem
If the Current Clique Algorithms are Optimal, so is Valiant's Parser
The CFG recognition problem is: given a context-free grammar
and a string of length , decide if can be obtained from
. This is the most basic parsing question and is a core computer
science problem. Valiant's parser from 1975 solves the problem in
time, where is the matrix multiplication
exponent. Dozens of parsing algorithms have been proposed over the years, yet
Valiant's upper bound remains unbeaten. The best combinatorial algorithms have
mildly subcubic complexity.
Lee (JACM'01) provided evidence that fast matrix multiplication is needed for
CFG parsing, and that very efficient and practical algorithms might be hard or
even impossible to obtain. Lee showed that any algorithm for a more general
parsing problem with running time can
be converted into a surprising subcubic algorithm for Boolean Matrix
Multiplication. Unfortunately, Lee's hardness result required that the grammar
size be . Nothing was known for the more relevant
case of constant size grammars.
In this work, we prove that any improvement on Valiant's algorithm, even for
constant size grammars, either in terms of runtime or by avoiding the
inefficiencies of fast matrix multiplication, would imply a breakthrough
algorithm for the -Clique problem: given a graph on nodes, decide if
there are that form a clique.
Besides classifying the complexity of a fundamental problem, our reduction
has led us to similar lower bounds for more modern and well-studied cubic time
problems for which faster algorithms are highly desirable in practice: RNA
Folding, a central problem in computational biology, and Dyck Language Edit
Distance, answering an open question of Saha (FOCS'14)
The DEPOSIT computer code: calculations of electron-loss cross sections for complex ions colliding with neutral atoms
A description of the DEPOSIT computer code is presented. The code is intended
to calculate total and m-fold electron-loss cross sections (m is the number of
ionized electrons) and the energy T(b) deposited to the projectile (positive or
negative ion) during a collision with a neutral atom at low and intermediate
collision energies as a function of the impact parameter b. The deposited
energy is calculated as a 3D-integral over the projectile coordinate space in
the classical energy-deposition model. Examples of the calculated deposited
energies, ionization probabilities and electron-loss cross sections are given
as well as the description of the input and output data.Comment: 11 pages, 3 figure
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