6,860 research outputs found

    N-fold integer programming in cubic time

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    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 nn-fold integer programming predating the present article runs in time O(ng(A)L)O(n^{g(A)}L) with LL the binary length of the numerical part of the input and g(A)g(A) the so-called Graver complexity of the bimatrix AA defining the system. In this article we provide a drastic improvement and establish an algorithm which runs in time O(n3L)O(n^3 L) having cubic dependency on nn regardless of the bimatrix AA. 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

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    The CFG recognition problem is: given a context-free grammar G\mathcal{G} and a string ww of length nn, decide if ww can be obtained from G\mathcal{G}. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in O(nω)O(n^{\omega}) time, where ω<2.373\omega<2.373 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 O(n3/log3n)O(n^3/\log^3{n}) 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 O(Gn3ε)O(|\mathcal{G}|\cdot n^{3-\varepsilon}) can be converted into a surprising subcubic algorithm for Boolean Matrix Multiplication. Unfortunately, Lee's hardness result required that the grammar size be G=Ω(n6)|\mathcal{G}|=\Omega(n^6). 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 kk-Clique problem: given a graph on nn nodes, decide if there are kk 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

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    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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