On one method of solving the partial covering problem

Cybernetics ◽  
1989 ◽  
Vol 25 (1) ◽  
pp. 122-125
Author(s):  
V. A. Roshchin ◽  
I. V. Sergienko
Keyword(s):  
Covering Problem ◽  
Partial Covering

scholarly journals Partial Covering of a Circle by 6 and 7 Congruent Circles

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2133
Author(s):  
Zsolt Gáspár ◽  
Tibor Tarnai ◽  
Krisztián Hincz
Keyword(s):  
Unit Circle ◽  
Numerical Solutions ◽  
Numerical Data ◽  
The Self ◽  
Covering Problem ◽  
Tabular Form ◽  
Geometrical Problem ◽  
Tensegrity Structure ◽  
Different Types ◽  
Partial Covering

Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by n congruent circles of given radius r, while r varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for n = 2 to 5. In the present paper, numerical solutions are given to this problem for n = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of n = 6 is very complicated, whilst the case n = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.

AN ALGORITHMIC FRAMEWORK FOR SOLVING GEOMETRIC COVERING PROBLEMS — WITH APPLICATIONS

2014 ◽  
Vol 25 (05) ◽  
pp. 623-639 ◽  
Author(s):  
TAHA GHASEMI ◽  
HOSSEIN GHASEMALIZADEH ◽  
MOHAMMADREZA RAZZAZI
Keyword(s):  
Approximation Algorithm ◽  
Approximation Schemes ◽  
Covering Problem ◽  
General View ◽  
Logarithmic Approximation ◽  
Covering Problems ◽  
Fixed Dimension ◽  
Minimum Number ◽  
Algorithmic Framework ◽  
Partial Covering

In geometric covering problems, the aim is to cover a set of given points using a minimum number of shapes with prescribed properties. These problems extend in many ways such as covering in the presence of obstacles and partial covering. Since in most cases these problems are NP-hard, researchers have developed a large number of approximation algorithms to solve them. In this paper, with a more general view on these approaches, we propose an algorithmic framework that gives an approximation algorithm for the covering problem at hand, provided that its prerequisites are satisfied. As the result, we present a general class of geometric covering problems for which one can obtain polynomial-time approximation schemes (PTASs). This class of problems involves covering point sets with bounded shapes in a space with fixed dimension. For a problem to be in this class, it must be possible to find a polynomial number of maximal shapes and it must be possible to cover all the points in a small box with a small number of shapes. Using this framework, we present two new PTASs for the problems of covering with disks in the presence of polygonal obstacles and covering with sectors (pie-shaped wedges). The first problem has never been addressed before and for the second problem, only a logarithmic approximation algorithm was known in general and a 9-approximation algorithm for wide angles.

A new reformulation approach for the generalized partial covering problem

2006 ◽  
Vol 53 (2) ◽  
pp. 170-179 ◽  
Author(s):  
Youngho Lee ◽  
Hanif D. Sherali ◽  
Ikhyun Kwon ◽  
Seongin Kim
Keyword(s):  
Covering Problem ◽  
Partial Covering

scholarly journals Q-Learnheuristics: Towards Data-Driven Balanced Metaheuristics

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1839
Author(s):  
Broderick Crawford ◽  
Ricardo Soto ◽  
José Lemus-Romani ◽  
Marcelo Becerra-Rozas ◽  
José M. Lanza-Gutiérrez ◽  
...  
Keyword(s):  
Combinatorial Problems ◽  
Set Covering ◽  
Covering Problem ◽  
Integration Framework ◽  
Q Learning ◽  
Whale Optimization ◽  
Sine Cosine Algorithm ◽  
Exploration Exploitation ◽  
Selection Of

One of the central issues that must be resolved for a metaheuristic optimization process to work well is the dilemma of the balance between exploration and exploitation. The metaheuristics (MH) that achieved this balance can be called balanced MH, where a Q-Learning (QL) integration framework was proposed for the selection of metaheuristic operators conducive to this balance, particularly the selection of binarization schemes when a continuous metaheuristic solves binary combinatorial problems. In this work the use of this framework is extended to other recent metaheuristics, demonstrating that the integration of QL in the selection of operators improves the exploration-exploitation balance. Specifically, the Whale Optimization Algorithm and the Sine-Cosine Algorithm are tested by solving the Set Covering Problem, showing statistical improvements in this balance and in the quality of the solutions.

scholarly journals An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 225
Author(s):  
José García ◽  
Gino Astorga ◽  
Víctor Yepes
Keyword(s):  
Transfer Functions ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Set Covering ◽  
Covering Problem ◽  
Perturbation Operator ◽  
K Nearest Neighbors ◽  
Combinatorial Optimization Problems ◽  
Box Plots ◽  
Metaheuristic Techniques

The optimization methods and, in particular, metaheuristics must be constantly improved to reduce execution times, improve the results, and thus be able to address broader instances. In particular, addressing combinatorial optimization problems is critical in the areas of operational research and engineering. In this work, a perturbation operator is proposed which uses the k-nearest neighbors technique, and this is studied with the aim of improving the diversification and intensification properties of metaheuristic algorithms in their binary version. Random operators are designed to study the contribution of the perturbation operator. To verify the proposal, large instances of the well-known set covering problem are studied. Box plots, convergence charts, and the Wilcoxon statistical test are used to determine the operator contribution. Furthermore, a comparison is made using metaheuristic techniques that use general binarization mechanisms such as transfer functions or db-scan as binarization methods. The results obtained indicate that the KNN perturbation operator improves significantly the results.

scholarly journals A fast $$(2 + \frac{2}{7})$$-approximation algorithm for capacitated cycle covering

2021 ◽  
Author(s):  
Vera Traub ◽  
Thorben Tröbst
Keyword(s):  
Vehicle Routing Problem ◽  
Covering Problem ◽  
Post Processing ◽  
Routing Problem ◽  
Cycle Cover ◽  
Disjoint Cycles ◽  
Processing Step ◽  
Number Of Cycles ◽  
Vertex Disjoint ◽  
Capacitated Vehicle

AbstractWe consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a $$2 + \epsilon $$ 2 + ϵ lower bound for the relaxation.

scholarly journals A Self-Adaptive Cuckoo Search Algorithm Using a Machine Learning Technique

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1840
Author(s):  
Nicolás Caselli ◽  
Ricardo Soto ◽  
Broderick Crawford ◽  
Sergio Valdivia ◽  
Rodrigo Olivares
Keyword(s):  
Machine Learning ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Cuckoo Search ◽  
Cuckoo Search Algorithm ◽  
Set Covering ◽  
Covering Problem ◽  
Parameter Configuration ◽  
Self Tuning ◽  
Problem Solvers

Metaheuristics are intelligent problem-solvers that have been very efficient in solving huge optimization problems for more than two decades. However, the main drawback of these solvers is the need for problem-dependent and complex parameter setting in order to reach good results. This paper presents a new cuckoo search algorithm able to self-adapt its configuration, particularly its population and the abandon probability. The self-tuning process is governed by using machine learning, where cluster analysis is employed to autonomously and properly compute the number of agents needed at each step of the solving process. The goal is to efficiently explore the space of possible solutions while alleviating human effort in parameter configuration. We illustrate interesting experimental results on the well-known set covering problem, where the proposed approach is able to compete against various state-of-the-art algorithms, achieving better results in one single run versus 20 different configurations. In addition, the result obtained is compared with similar hybrid bio-inspired algorithms illustrating interesting results for this proposal.

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