Special Issue "Discrete Optimization: Theory, Algorithms, and Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 28 February 2019

Special Issue Editor

Guest Editor
Prof. Dr. Frank Werner

Faculty of Mathematics, Otto-von-Guericke-University, P.O. Box 4120, D-39016 Magdeburg, Germany
Website | E-Mail
Interests: discrete optimization; operations research; scheduling; graph theory; manufacturing systems

Special Issue Information

Dear Colleagues,

We invite you to submit your latest research in the area of discrete optimization to this Special Issue, “Discrete Optimization: Theory, Algorithms, and Applications” in the journal Mathematics. We are looking for new and innovative approaches for solving discrete optimization problems exactly or approximately. High-quality papers are solicited to address both theoretical and practical issues of discrete optimization. Submissions are welcome presenting new theoretical results, structural investigations, new models and algorithmic approaches as well new applications of discrete optimization problems. Potential topics include, but are not limited to, integer programming, combinatorial optimization, graph-theoretic problems, matroids, scheduling, and logistics.   

Prof. Dr. Frank Werner
Guest Editor

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Keywords

  • Nonlinear and Linear Integer Programming
  • Optimization on Graphs and Networks
  • Greedy Algorithms, Matroids and Submodular Functions
  • Polyhedral Combinatorics
  • Combinatorial Optimization
  • Scheduling
  • Robust Discrete Optimization
  • Optimization under Uncertainty
  • Computational Complexity
  • Branch and Bound, Cutting-Plane Methods, Dynamic Programming
  • Approximation and Randomized Algorithms
  • Metaheuristics, Matheuristics
  • Interior Point Methods
  • Decomposition Methods

Published Papers (16 papers)

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Research

Open AccessArticle Further Results on the Resistance-Harary Index of Unicyclic Graphs
Mathematics 2019, 7(2), 201; https://doi.org/10.3390/math7020201 (registering DOI)
Received: 20 December 2018 / Revised: 14 February 2019 / Accepted: 14 February 2019 / Published: 20 February 2019
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Abstract
The Resistance-Harary index of a connected graph G is defined as RH(G)={u,v}V(G)1r(u,v), where r(u,v) [...] Read more.
The Resistance-Harary index of a connected graph G is defined as R H ( G ) = { u , v } V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Fractional Metric Dimension of Generalized Jahangir Graph
Mathematics 2019, 7(1), 100; https://doi.org/10.3390/math7010100
Received: 30 November 2018 / Revised: 7 January 2019 / Accepted: 9 January 2019 / Published: 18 January 2019
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Abstract
Arumugam and Mathew [Discret. Math. 2012, 312, 1584–1590] introduced the notion of fractional metric dimension of a connected graph. In this paper, a combinatorial technique is devised to compute it. In addition, using this technique the fractional metric dimension of [...] Read more.
Arumugam and Mathew [Discret. Math. 2012, 312, 1584–1590] introduced the notion of fractional metric dimension of a connected graph. In this paper, a combinatorial technique is devised to compute it. In addition, using this technique the fractional metric dimension of the generalized Jahangir graph J m , k is computed for k 0 and m = 5 . Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Fault-Tolerant Resolvability and Extremal Structures of Graphs
Mathematics 2019, 7(1), 78; https://doi.org/10.3390/math7010078
Received: 25 November 2018 / Revised: 31 December 2018 / Accepted: 10 January 2019 / Published: 14 January 2019
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Abstract
In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n1, and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, [...] Read more.
In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Distance and Adjacency Energies of Multi-Level Wheel Networks
Mathematics 2019, 7(1), 43; https://doi.org/10.3390/math7010043
Received: 11 December 2018 / Revised: 25 December 2018 / Accepted: 27 December 2018 / Published: 4 January 2019
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Abstract
Energies of molecular graphs have various applications in chemistry, polymerization, computer networking and pharmacy. In this paper, we give general closed forms of distance and adjacency energies of generalized wheel networks Wn,m. Consequently, we give these results for classical [...] Read more.
Energies of molecular graphs have various applications in chemistry, polymerization, computer networking and pharmacy. In this paper, we give general closed forms of distance and adjacency energies of generalized wheel networks W n , m . Consequently, we give these results for classical wheel graphs. We also give pictorial dependencies of energies on the involved parameters m 3 and n . Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Construction Algorithm for Zero Divisor Graphs of Finite Commutative Rings and Their Vertex-Based Eccentric Topological Indices
Mathematics 2018, 6(12), 301; https://doi.org/10.3390/math6120301
Received: 26 October 2018 / Revised: 30 November 2018 / Accepted: 2 December 2018 / Published: 4 December 2018
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Abstract
Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. [...] Read more.
Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this paper we discuss eccentric topological indices of zero divisor graphs of commutative rings Z p 1 p 2 × Z q , where p 1 , p 2 , and q are primes. To enhance the importance of these indices a construction algorithm is also devised for zero divisor graphs of commutative rings Z p 1 p 2 × Z q . Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
Open AccessArticle Resistance Distance in H-Join of Graphs G1,G2,,Gk
Mathematics 2018, 6(12), 283; https://doi.org/10.3390/math6120283
Received: 11 October 2018 / Revised: 18 November 2018 / Accepted: 21 November 2018 / Published: 26 November 2018
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Abstract
In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G1,G2, [...] Read more.
In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G 1 , G 2 , , G k . Recall that H is an arbitrary graph with V ( H ) = { 1 , 2 , , k } , and G 1 , G 2 , , G k are disjoint graphs. Then, the H-join of graphs G 1 , G 2 , , G k , denoted by H { G 1 , G 2 , , G k } , is a graph formed by taking G 1 , G 2 , , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of H { G 1 , G 2 , , G k } , and then give a { 1 } -inverse L ( H { G 1 , G 2 , , G k } ) { 1 } or group inverse L ( H { G 1 , G 2 , , G k } ) # of L ( H { G 1 , G 2 , , G k } ) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in H { G 1 , G 2 , , G k } . In addition, some applications are presented in this paper. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness
Mathematics 2018, 6(11), 271; https://doi.org/10.3390/math6110271
Received: 12 October 2018 / Revised: 16 November 2018 / Accepted: 16 November 2018 / Published: 21 November 2018
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Abstract
The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the [...] Read more.
The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than m . Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
Open AccessArticle Scheduling and Planning in Service Systems with Goal Programming: Literature Review
Mathematics 2018, 6(11), 265; https://doi.org/10.3390/math6110265
Received: 1 November 2018 / Revised: 13 November 2018 / Accepted: 14 November 2018 / Published: 19 November 2018
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Abstract
Background: People want to be able to evaluate different kinds of information in a good way. There are various methods that they develop in such situations. Among the optimization methods, the goal programming method is often used when there are multiple objectives that [...] Read more.
Background: People want to be able to evaluate different kinds of information in a good way. There are various methods that they develop in such situations. Among the optimization methods, the goal programming method is often used when there are multiple objectives that decision makers want to accomplish. Because scheduling and planning problems have multiple objectives that are desired to be achieved, the goal programming method helps the researcher in contradictory situations between these goals. Methods: This study includes, examines, and analyzes recent research on service scheduling and planning. In the literature, service scheduling and planning studies have been examined using goal programming method from past to today. Results: The studies are detailed according to the type of goal programming, according to scheduling types, the purpose used in the studies, and the methods integrated with the goal programming. There are 142 studies in Emerald, Science Direct, Jstor, Springer, Taylor and Francis, Google Scholar, etc. databases that are examined in detail. For readers, diversification has been made to facilitate the identification of these studies and a detailed overview has been presented. Conclusion: As a result of the study, studies with the goal programming in the literature have been seen. The readers’ perspectives for planning and scheduling are discussed. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
Open AccessArticle Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph
Mathematics 2018, 6(11), 243; https://doi.org/10.3390/math6110243
Received: 23 September 2018 / Revised: 27 October 2018 / Accepted: 29 October 2018 / Published: 7 November 2018
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Abstract
Consider an undirected and connected graph G=(VG,EG), where VG and EG represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly [...] Read more.
Consider an undirected and connected graph G = ( V G , E G ) , where V G and E G represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly resolving sets is based on the distances of edges in a graph. In this paper, we find the edge version of metric dimension and doubly resolving sets for the necklace graph. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Maximizing and Minimizing Multiplicative Zagreb Indices of Graphs Subject to Given Number of Cut Edges
Mathematics 2018, 6(11), 227; https://doi.org/10.3390/math6110227
Received: 21 September 2018 / Revised: 19 October 2018 / Accepted: 22 October 2018 / Published: 29 October 2018
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Abstract
Given a (molecular) graph, the first multiplicative Zagreb index Π1 is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index Π2 is expressed as the product of endvertex degree of each [...] Read more.
Given a (molecular) graph, the first multiplicative Zagreb index Π 1 is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index Π 2 is expressed as the product of endvertex degree of each edge over all edges. We consider a set of graphs G n , k having n vertices and k cut edges, and explore the graphs subject to a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs in G n , k are provided. We also provide these graphs with the largest and smallest Π 1 ( G ) and Π 2 ( G ) in G n , k . Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)
Mathematics 2018, 6(10), 206; https://doi.org/10.3390/math6100206
Received: 11 September 2018 / Revised: 9 October 2018 / Accepted: 10 October 2018 / Published: 16 October 2018
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Abstract
A double Roman dominating function (DRDF) f on a given graph G is a mapping from V(G) to {0,1,2,3} in such a way that a vertex u for which f(u [...] Read more.
A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) to { 0 , 1 , 2 , 3 } in such a way that a vertex u for which f ( u ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f ( u ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = u V ( G ) f ( u ) . The minimum weight of a DRDF on a graph G is called the double Roman domination number γ d R ( G ) of G. In this paper, we determine the exact value of the double Roman domination number of the generalized Petersen graphs P ( n , 2 ) by using a discharging approach. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product
Mathematics 2018, 6(10), 191; https://doi.org/10.3390/math6100191
Received: 25 July 2018 / Revised: 24 September 2018 / Accepted: 26 September 2018 / Published: 8 October 2018
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Abstract
Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex [...] Read more.
Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Optimizing Three-Dimensional Constrained Ordered Weighted Averaging Aggregation Problem with Bounded Variables
Mathematics 2018, 6(9), 172; https://doi.org/10.3390/math6090172
Received: 3 September 2018 / Revised: 13 September 2018 / Accepted: 14 September 2018 / Published: 19 September 2018
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Abstract
A single constrained ordered weighted averaging aggregation (COWA) problem is of considerable importance in many disciplines. Two models are considered: the maximization COWA problem with lower bounded variables and the minimization COWA problem with upper bounded variables. For a three-dimensional case of these [...] Read more.
A single constrained ordered weighted averaging aggregation (COWA) problem is of considerable importance in many disciplines. Two models are considered: the maximization COWA problem with lower bounded variables and the minimization COWA problem with upper bounded variables. For a three-dimensional case of these models, we present the explicitly optimal solutions theoretically and empirically. The bounds and weights can affect the optimal solution of the three-dimensional COWA problem with bounded variables. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Computing The Irregularity Strength of Planar Graphs
Mathematics 2018, 6(9), 150; https://doi.org/10.3390/math6090150
Received: 23 July 2018 / Revised: 25 August 2018 / Accepted: 27 August 2018 / Published: 30 August 2018
Cited by 1 | PDF Full-text (281 KB) | HTML Full-text | XML Full-text
Abstract
The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base [...] Read more.
The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management. In this paper, we discuss the totally irregular total k labeling of three planar graphs. If such labeling exists for minimum value of a positive integer k, then this labeling is called totally irregular total k labeling and k is known as the total irregularity strength of a graph G. More preciously, we determine the exact value of the total irregularity strength of three planar graphs. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Computing Topological Indices and Polynomials for Line Graphs
Mathematics 2018, 6(8), 137; https://doi.org/10.3390/math6080137
Received: 16 July 2018 / Revised: 31 July 2018 / Accepted: 5 August 2018 / Published: 10 August 2018
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Abstract
A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the [...] Read more.
A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Eccentricity Based Topological Indices of an Oxide Network
Mathematics 2018, 6(7), 126; https://doi.org/10.3390/math6070126
Received: 5 June 2018 / Revised: 8 July 2018 / Accepted: 11 July 2018 / Published: 18 July 2018
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Abstract
Graph theory has much great advances in the field of mathematical chemistry. Chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are [...] Read more.
Graph theory has much great advances in the field of mathematical chemistry. Chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of an oxide network and compute the total eccentricity, average eccentricity, eccentricity based Zagreb indices, atom-bond connectivity (ABC) index and geometric arithmetic index of an oxide network. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies. Full article
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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