Feature Papers in Randomized, Online and Approximation Algorithms

A topical collection in Algorithms (ISSN 1999-4893). This collection belongs to the section "Randomized, Online, and Approximation Algorithms".

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Editor


E-Mail Website
Collection Editor
Faculty of Mathematics, Otto-von-Guericke-University, P.O. Box 4120, D-39016 Magdeburg, Germany
Interests: scheduling; development of exact and approximate algorithms; stability investigations; discrete optimization; scheduling with interval processing times; complex investigations for scheduling problems; train scheduling; graph theory; logistics; supply chains; packing; simulation; applications
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Topical Collection Information

Dear Colleagues,

Many problems in different fields of research and application are so complex that one can find only approximate solutions within reasonable computational time, or they require an algorithm which makes decisions online on the basis of incomplete information. This means that either the search space is too large and complex to efficiently find an optimal solution, or the search space is not completely known. There has been substantial progress in the development of algorithms for such problems over the past decades.

This topical collection is dedicated to the presentation of new and innovative results in the field of the design and analysis of randomized, online, or approximation algorithms. This selection looks both for theoretical results and applications in the real world. Survey papers highlighting the most recent advances and trends in this field are also welcome. There are no restrictions regarding the length of a submission.

Prof. Dr. Frank Werner
Collection Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the collection website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Algorithms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical programming
  • operations research
  • production planning, scheduling, logistics, and transport
  • combinatorial optimization
  • discrete mathematics, graph theory, and networks
  • machine learning
  • multi-criteria decision making
  • design and analysis of algorithms
  • distributed and parallel algorithms
  • approximation algorithms
  • parametrized approximation
  • metaheuristics and matheuristics
  • randomized algorithms
  • online algorithms and competitive analysis
  • new applications in real-world problems

Published Papers (3 papers)

2023

22 pages, 1204 KiB  
Article
An Algorithm for Construction of the Asymptotic Approximation of a Stable Stationary Solution to a Diffusion Equation System with a Discontinuous Source Function
by Nikolay Nefedov, Bogdan Tishchenko and Natalia Levashova
Algorithms 2023, 16(8), 359; https://doi.org/10.3390/a16080359 - 26 Jul 2023
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Abstract
An algorithm is presented for the construction of an asymptotic approximation of a stable stationary solution to a diffusion equation system in a two-dimensional domain with a smooth boundary and a source function that is discontinuous along some smooth curve lying entirely inside [...] Read more.
An algorithm is presented for the construction of an asymptotic approximation of a stable stationary solution to a diffusion equation system in a two-dimensional domain with a smooth boundary and a source function that is discontinuous along some smooth curve lying entirely inside the domain. Each of the equations contains a small parameter as a factor in front of the Laplace operator, and as a result, the system is singularly perturbed. In the vicinity of the curve, the solution of the system has a large gradient. Such a problem statement is used in the model of urban development in metropolitan areas. The discontinuity curves in this model are the boundaries of urban biocenoses or large water pools, which prevent the spread of urban development. The small parameter is the ratio of the city’s outskirts linear size to the whole metropolis linear size. The algorithm includes the construction of an asymptotic approximation to a solution with a large gradient at the media interface as well as the steps for obtaining the existence conditions. To prove the existence and stability theorems, we use the upper and lower solutions, which are constructed as modifications of the asymptotic approximation to the solution. The latter is constructed using the Vasil’yeva algorithm as an expansion of a small parameter exponent. Full article
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24 pages, 604 KiB  
Article
ω-Circulant Matrices: A Selection of Modern Applications from Preconditioning of Approximated PDEs to Subdivision Schemes
by Rafael Díaz Fuentes, Stefano Serra-Capizzano and Rosita Luisa Sormani
Algorithms 2023, 16(7), 328; https://doi.org/10.3390/a16070328 - 08 Jul 2023
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Abstract
It is well known that ω-circulant matrices with ω0 can be simultaneously diagonalized by a transform matrix, which can be factored as the product of a diagonal matrix, depending on ω, and of the unitary matrix Fn associated [...] Read more.
It is well known that ω-circulant matrices with ω0 can be simultaneously diagonalized by a transform matrix, which can be factored as the product of a diagonal matrix, depending on ω, and of the unitary matrix Fn associated to the Fast Fourier Transform. Hence, all the sets of ω-circulants form algebras whose computational power, in terms of complexity, is the same as the classical circulants with ω=1. However, stability is a delicate issue, since the condition number of the transform is equal to that of the diagonal part, tending to max{|ω|,|ω|1}. For ω=0, the set of related matrices is still an algebra, which is the algebra of lower triangular matrices, but they do not admit a common transform since most of them (all except the multiples of the identity) are non-diagonalizable. In the present work, we review two modern applications, ranging from parallel computing in preconditioning of PDE approximations to algorithms for subdivision schemes, and we emphasize the role of such algebra. For the two problems, few numerical tests are conducted and critically discussed and the related conclusions are drawn. Full article
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14 pages, 534 KiB  
Communication
Algorithm for Approximate Solving of a Nonlinear Boundary Value Problem for Generalized Proportional Caputo Fractional Differential Equations
by Angel Golev, Snezhana Hristova and Asen Rahnev
Algorithms 2023, 16(6), 272; https://doi.org/10.3390/a16060272 - 29 May 2023
Viewed by 917
Abstract
In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on [...] Read more.
In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations. Full article
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