Some Problems in Graph Theory and Scheduling

In this paper, we first establish a Multi-Relation Granular Computing model by a given graph, and point out that all the reducts of the constructed Multi-Relation Granular Computing model are exactly all the minimal vertex covers of the corresponding graph. Thus, the vertex cover problem in graph theory can be converted to the knowledge reduction problem in rough set theory. Based on the conversion, we then introduce methods for dealing with the knowledge reduction problem to solve the vertex cover problem. In particular, we introduce a kind of method called the heuristic reduction algorithm based on entropy.

There will be a lot of NP- complete problems in graph theory and optimization process, as the most important problem in scientific engineering computing, now it is generally used genetic algorithm to solve. Therefore, this article will mainly study the basic theory of genetic algorithms and graph theory, and put forward the optimization algorithm in graph theory vertex cover problem in the specific application.

In the field of research on Deoxyribonucleic acid (DNA), graph theory can be applied to model the structure of a DNA molecule. In particular, the shortest path problem in graph theory can be used to identify the shortest path between vertices of a graph. Hence, it is possible to apply the shortest path problem for minimization of a DNA string so that the time taken for computation of genome assembly can be reduced. This paper presents a method to represent a DNA string graphically where the shortest path is calculated for the graph generated from the DNA string, and the shortest path is then used to minimize the DNA string. In this research, a DNA string is presented in graphical form by using base pairs of length two as the vertices where the initial bases used are Adenine (A) and Guanine (G) which are the main bases in purine. The number of base pairs between adjacent vertices in the DNA string is represented by the edges. The graph is then reduced by following a given set of rules where the shortest path is calculated for all start and end vertices of the reduced graph. Next, the simplification of the graph is done based on the shortest paths obtained by removing all the untraversed paths where the Euler path for the simplified graph is used to form a minimized DNA string. The result shows that a minimized DNA string can be obtained by simplifying the graph of the DNA string using the shortest path problem

A novel class of switching networks has been proposed to remove the time and space bottlenecks in conventional RAM-controlled switching architectures. The proposed networks possess many desirable characteristics for high-speed electronic and photonic switching systems, such as tolerance of faults, O(log/sub 2/N) stages between each inlet-outlet pair, self-routing capability, easy path hunt and easy fault diagnosis. Graph theory provides the basis for the proposed networks. It is shown that the key problem in the construction of this class of networks is related to the coloring problem in graph theory. >

Motivated by the celebrated paper of Hooker (J Heuristics 1(1): 33–42, 1995) published in the first issue of this journal, and by the relative lack of progress of both approximation algorithms and fixed-parameter algorithms for the classical decision and optimization problems related to covering edges by vertices, we aimed at developing an approach centered in augmenting our intuition about what is indeed needed. We present a case study of a novel design methodology by which algorithm weaknesses will be identified by computer-based and fixed-parameter tractable algorithmic challenges on their performance. Comprehensive benchmarkings on all instances of small size then become an integral part of the design process. Subsequent analyses of cases where human intuition “fails”, supported by computational testing, will then lead to the development of new methods by avoiding the traps of relying only on human perspicacity and ultimately will improve the quality of the results. Consequently, the computer-aided design process is seen as a tool to augment human intuition. It aims at accelerating and foster theory development in areas such as graph theory and combinatorial optimization since some safe reduction rules for pre-processing can be mathematically proved via theorems. This approach can also lead to the generation of new interesting heuristics. We test our ideas with a fundamental problem in graph theory that has attracted the attention of many researchers over decades, but for which seems it seems to be that a certain stagnation has occurred. The lessons learned are certainly beneficial, suggesting that we can bridge the increasing gap between theory and practice by a more concerted approach that would fuel human imagination from a data-driven discovery perspective.

The matching problem is one of the central problems in graph theory as well as in the theory of algorithms and their applications. This book will provide the reader with a comprehensive and straightforward introduction to the basic methods of designing efficient parallel algorithms for graph matching problems. The text is written for students at the beginning graduate level. The exposition is mostly self-contained and example-driven. Prerequisites have been kept to a minimum by including relevant background material. The book contains full details of several new techniques and should also be of interest to research workers in computer science, operations research, discrete mathematics, and electrical engineering. The main theoretical tools are combined into three independent chapters, devoted to combinatorial tools, probabilistic tools, and algebraic tools. One of the main goals of the book is to bring together these three approaches and highlight how their combination works in the development of efficient parallel algorithms. The reader will be provided with a simple and transparent presentation of a variety of interesting algorithms, including many examples and illustrations. The combination of different approaches makes the matching problem and its applications an attractive and fascinating subject. It is hoped that the book represents a meeting point of interesting algorithmic techniques and opens up new algebraic and geometric areas. Marek Karpinski is Chair Professor of Computer Science at the University of Bonn. Wojciech Rytter is Professor of Computer Science at the University of Warsaw and at the University of Liverpool.

In this work, we study the availability-aware survivable virtual network embedding (A-SVNE) problem in optical interdatacenter networks that use wavelength-division multiplexing. With A-SVNE, we try to satisfy the availability requirement of each virtual component (i.e., a virtual link or a virtual node) in a virtual network. We first analyze the availability of a virtual component based on the availabilities of the substrate link(s) and node(s). Then, we formulate an integer linear programming model for the A-SVNE problem and propose several time-efficient heuristics. Specifically, we design two node mapping strategies: one is sequential selection using efficient weights defined by the availability information, while the other uses auxiliary graphs to transform the problem into a classical problem in graph theory, i.e., the maximum-weight maximum clique. Finally, we use extensive simulations to compare the proposed A-SVNE algorithms with existing ones in terms of the blocking probability, availability gap, and penalty due to service-level agreement violations, and the results indicate that our algorithms perform better.

SummaryHybrid networks, comprising a conventional cellular network overlaid with Device‐to‐Device (D2D), offer efficient way to improve system throughput. In this paper, a novel orthogonal frequency‐division multiple access channel‐assignment method is proposed for hybrid network. The proposed approach is optimal in terms of throughput and is subjected to a sensible QoS requirement, which guarantees that macrocell and D2D achieve a prescribed data rate and outage probability, respectively. Our solution consists of two phases. In the first phase, the minimum sub‐channels are allocated to the macrocell to satisfy their data rate requirements. This problem is mapped to the 0‐1 Knapsack Problem and solved by integer programming based Lagrange dual approach. In the second phase, the redundant sub‐channels are allocated to D2D pairs to maximize the throughput of D2D networks. An interference management scheme is proposed to guarantee the outage probability of D2D communications. A cluster is taken as the unit for frequency reuse among D2D pairs. The problem of clustering is mapped to the MAXk‐CUT problem in graph theory and is solved by graph‐based heuristic algorithm. Extensive simulations demonstrate the superior performance of the proposed solution compared with the existing scheme. Copyright © 2014 John Wiley & Sons, Ltd.

Graph matching is one of the most important problems in graph theory and combinatorial optimization, with many applications in various domains. Although meta-heuristic algorithms have had good performance on many NP-Hard and NP-Complete problems, but for graph matching problem, there were not reported superior solutions by these sort of algorithms. The reason of this inefficiency has not been investigated yet. In this paper it has been shown that Simulated Annealing (SA) as an instance of a meta-heuristic method is unlikely to be even close to the optimal solution for this problem. Mathematical and experimental results showed that the chance to reach to a partial solution, is very low, even for small number of true matches. In addition to theoretical discussion, the experimental results also verified our idea; for example, in two sample graphs with $10000$ vertices, the probability of reaching to a solution with at least three correct matches is about $0.02$ with simulated annealing.

Simplicial Decompositions of Graphs: a Survey of Applications

My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems.

A Characterization of Perfect Matrices

The authors present some analytical results for the Potts model, using the symmetry group (acting on its parameters) generated by the inverse relation and other symmetries of this model. In particular, they find the critical manifolds and study the relationship between this group and the Lee-Yang singularities in the complex plane. For those cases where the symmetry group is finite, they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials.

Motivated by an application in controllability we consider maximum matchings in random bipartite graphs G = (A, B). First we analyse Karp–Sipser's algorithm to determine the asymptotic size of maximum matchings in random bipartite graphs with a fixed degree distribution. We then allow an adversary to delete one edge adjacent to every vertex in A in the more restricted model where each vertex in A chooses d neighbours uniformly at random from B.

Electron track reconstruction and improved modulation for photoelectric X-ray polarimetry