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Mathematics
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Dynamic Programming
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Dynamic Programming

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 March 2025) | Viewed by 7002

Special Issue Editor

Dr. Abdullah N. Arslan

E-Mail Website
Guest Editor
Department of Computer Science, Texas A&M University-Commerce, Commerce, TX 75428, USA
Interests: dynamic programming; fractional programming; graph algorithms; string algorithms

Special Issue Information

Dear Colleagues,

Dynamic programming is an algorithm design technique suitable for solving certain optimization problems. A major characteristic of a dynamic programming solution is that sub-problems are solved in an order of increasing problem size, by which solving the next sub-problem makes use of recorded solutions of sub-problems encountered earlier. This avoids solving the same sub-problems repeatedly. Dynamic programming yields efficient algorithms for many optimization problems on graphs (e.g., all-pairs shortest paths), and patterns (e.g., edit distance, sequence alignment). Dynamic-programming-based algorithms have applications in a wide array of research areas, including computational biology, computational finance, computational economics, computational intelligence, machine learning, artificial intelligence, operations research, business analytics, data analysis, and machine learning. This Special Issue on dynamic programming aims to bring together articles that present novel ideas and new solutions that use dynamic programming for computational problems. Reviews that provide new focus or new perspectives for dynamic programming algorithms and applications are also welcome.

Dr. Abdullah N. Arslan
Guest Editor

Manuscript Submission Information

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Keywords

  • algorithm
  • optimization
  • computational problem
  • mathematical programming
  • bottom-up approach
  • iterative method
  • table lookup

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Published Papers (4 papers)

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Research

15 pages, 1183 KiB  
Article
Identifying All Matches of a Rigid Object in an Input Image Using Visible Triangles
by Abdullah N. Arslan
Mathematics 2025, 13(6), 925; https://doi.org/10.3390/math13060925 - 11 Mar 2025
Viewed by 393
Abstract
It has been suggested that for objects identifiable by their corners, every triangle formed by these corner points can serve as a reference for detecting other corner points. This approach enables effective rigid object detection, including partial matches. However, when there are many [...] Read more.
It has been suggested that for objects identifiable by their corners, every triangle formed by these corner points can serve as a reference for detecting other corner points. This approach enables effective rigid object detection, including partial matches. However, when there are many corner points, the implementation becomes impractical due to excessive memory requirements. To overcome this, we propose a new algorithm that leverages Delaunay triangulation, considering only the triangles generated by the Delaunay triangulation to reduce the complexity of the original approach. Our algorithm is significantly faster and requires significantly less memory, offering a viable solution for large problem instances. Moreover, it excels at identifying all matches of a queried object in an image when visible triangles of the object are present. A triangle formed by an object’s vertices is considered visible if a matching triangle is detected, and no vertices from any other object lie within its circumcircle. Recent AI-based methods have revolutionized rigid object matching, providing impressive accuracy with deep learning techniques. However, these methods require extensive training and specialized hardware like GPUs. In contrast, our approach requires no training or specialized hardware, making it a lightweight and efficient solution that maintains strong matching capabilities without the overhead of AI-based methods. Our study of the geometric features, combined with Delaunay triangulation, offers new mathematical insights. Full article
(This article belongs to the Special Issue Dynamic Programming)
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22 pages, 586 KiB  
Article
A New Alternating Suboptimal Dynamic Programming Algorithm with Applications for Feature Selection
by David Podgorelec, Borut Žalik, Domen Mongus and Dino Vlahek
Mathematics 2024, 12(13), 1987; https://doi.org/10.3390/math12131987 - 27 Jun 2024
Cited by 2 | Viewed by 1218
Abstract
Feature selection is predominantly used in machine learning tasks, such as classification, regression, and clustering. It selects a subset of features (relevant attributes of data points) from a larger set that contributes as optimally as possible to the informativeness of the model. There [...] Read more.
Feature selection is predominantly used in machine learning tasks, such as classification, regression, and clustering. It selects a subset of features (relevant attributes of data points) from a larger set that contributes as optimally as possible to the informativeness of the model. There are exponentially many subsets of a given set, and thus, the exhaustive search approach is only practical for problems with at most a few dozen features. In the past, there have been attempts to reduce the search space using dynamic programming. However, models that consider similarity in pairs of features alongside the quality of individual features do not provide the required optimal substructure. As a result, algorithms, which we will call suboptimal dynamic programming algorithms, find a solution that may deviate significantly from the optimal one. In this paper, we propose an iterative dynamic programming algorithm, which invertsthe order of feature processing in each iteration. Such an alternating approach allows for improving the optimization function by using the score from the previous iteration to estimate the contribution of unprocessed features. The iterative process is proven to converge and terminates when the solution does not change in three successive iterations or when the number of iterations reaches the threshold. Results in more than 95% of tests align with those of the exhaustive search approach, being competitive and often superior to the reference greedy approach. Validation was carried out by comparing the scores of output feature subsets and examining the accuracy of different classifiers learned on these features across nine real-world applications, considering different scenarios with various numbers of features and samples. In the context of feature selection, the proposed algorithm can be characterized as a robust filter method that can improve machine learning models regardless of dataset size. However, we expect that the idea of alternating suboptimal optimization will soon be generalized to tasks beyond feature selection. Full article
(This article belongs to the Special Issue Dynamic Programming)
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22 pages, 860 KiB  
Article
A Dynamic Programming Approach to the Collision Avoidance of Autonomous Ships
by Raphael Zaccone
Mathematics 2024, 12(10), 1546; https://doi.org/10.3390/math12101546 - 15 May 2024
Cited by 7 | Viewed by 1988
Abstract
The advancement of autonomous capabilities in maritime navigation has gained significant attention, with a trajectory moving from decision support systems to full autonomy. This push towards autonomy has led to extensive research focusing on collision avoidance, a critical aspect of safe navigation. Among [...] Read more.
The advancement of autonomous capabilities in maritime navigation has gained significant attention, with a trajectory moving from decision support systems to full autonomy. This push towards autonomy has led to extensive research focusing on collision avoidance, a critical aspect of safe navigation. Among the various possible approaches, dynamic programming is a promising tool for optimizing collision avoidance maneuvers. This paper presents a DP formulation for the collision avoidance of autonomous vessels. We set up the problem framework, formulate it as a multi-stage decision process, define cost functions and constraints focusing on the actual requirements a marine maneuver must comply with, and propose a solution algorithm leveraging parallel computing. Additionally, we present a greedy approximation to reduce algorithm complexity. We put the proposed algorithms to the test in realistic navigation scenarios and also develop an extensive test on a large set of randomly generated scenarios, comparing them with the RRT* algorithm using performance metrics proposed in the literature. The results show the potential benefits of an autonomous navigation or decision support framework. Full article
(This article belongs to the Special Issue Dynamic Programming)
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28 pages, 3084 KiB  
Article
Polynomial-Time Constrained Message Passing for Exact MAP Inference on Discrete Models with Global Dependencies
by Alexander Bauer, Shinichi Nakajima and Klaus-Robert Müller
Mathematics 2023, 11(12), 2628; https://doi.org/10.3390/math11122628 - 8 Jun 2023
Cited by 2 | Viewed by 1628
Abstract
Considering the worst-case scenario, the junction-tree algorithm remains the most general solution for exact MAP inference with polynomial run-time guarantees. Unfortunately, its main tractability assumption requires the treewidth of a corresponding MRF to be bounded, strongly limiting the range of admissible applications. In [...] Read more.
Considering the worst-case scenario, the junction-tree algorithm remains the most general solution for exact MAP inference with polynomial run-time guarantees. Unfortunately, its main tractability assumption requires the treewidth of a corresponding MRF to be bounded, strongly limiting the range of admissible applications. In fact, many practical problems in the area of structured prediction require modeling global dependencies by either directly introducing global factors or enforcing global constraints on the prediction variables. However, this always results in a fully-connected graph, making exact inferences by means of this algorithm intractable. Previous works focusing on the problem of loss-augmented inference have demonstrated how efficient inference can be performed on models with specific global factors representing non-decomposable loss functions within the training regime of SSVMs. Making the observation that the same fundamental idea can be applied to solve a broader class of computational problems, in this paper, we adjust the framework for an efficient exact inference to allow much finer interactions between the energy of the core model and the sufficient statistics of the global terms. As a result, we greatly increase the range of admissible applications and strongly improve upon the theoretical guarantees of computational efficiency. We illustrate the applicability of our method in several use cases, including one that is not covered by the previous problem formulation. Furthermore, we propose a new graph transformation technique via node cloning, which ensures a polynomial run-time for solving our target problem. In particular, the overall computational complexity of our constrained message-passing algorithm depends only on form-independent quantities such as the treewidth of a corresponding graph (without global connections) and image size of the sufficient statistics of the global terms. Full article
(This article belongs to the Special Issue Dynamic Programming)
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