A rough set method for the vertex cover problem in graph theory

In this letter, we study the maximum access problem (MAP) for uplink networks with non-orthogonal multiple access technique. Specifically, the MAP is formulated as a mixed integer programming problem with admission control, user clustering, channel assignment, and power control taken into account. To deal with this challenging problem, we first recast it as the maximum independent set (MIS) problem in graph theory. Then, we propose an efficient algorithm to solve this MIS problem in a low complexity manner. Finally, simulation results indicate that our algorithm can greatly improve the number of supported users compared to other schemes. The performance of our algorithm under different channel conditions and non-uniform distributions of users is also investigated.

Independent domination in finitely defined classes of graphs: Polynomial algorithms

We present parallel approximation schemes for a large class of problems in logic and graph theory, when these problems are restricted to planar graphs and δ-near-planar graphs. Our results are based on the positive use of L-reductions, and the decomposition of a given graph into subgraphs for which the given problem can be solved optimally in polynomial time. The problems considered include MAX 3SAT, MAX SAT(S), maximum independent set, and minimum dominating set. Our NC-approximation schemes exhibit the same time versus performance tradeoff as those of Baker [Ba83]. For δ-near-planar graphs, this is the first time that approximation schemes of any kind have been obtained for these problems.

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.

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.

To monitor an electric power system by placing as few phase measurement units (PMUs) as possible is closely related to the famous vertex cover problem and domination problem in graph theory. A set P is a power dominating set (PDS) of a graph G = (V, E), if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number γp(G). In this paper, we determine the upper bounds of power domination number of generalized undirected de Bruijn graphs and generalized undirected Kautz graphs.

We consider the problem of constructing a graph having some given number of independent sets. The bounds are obtained for the number of vertices in bipartite graphs with the prescribed number of independent sets and the number of inclusion maximal independent sets.

Polar graphs and maximal independent sets

In this dissertation, we present three results related to combinatorial algorithms in graph theory and scheduling, both of which are important subjects in the area of discrete mathematics and theoretical computer science. In graph theory, a graph is a set of vertices and edges, where each edge is a pair of vertices. A coloring of a graph is a function that assigns each vertex a color such that no two adjacent vertices share the same color. The first two results are related to coloring graphs belonging to specific classes. In scheduling problems, we are interested in how to efficiently schedule a set of jobs on machines. The last result is related to a scheduling problem in an environment where there is uncertainty on the number of machines. The first result of this thesis is a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable. This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002. Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1, 2, 3}, and gives an explicit coloring if one exists. This is joint work with Flavia Bonomo, Maria Chundnovsky, Peter Maceli, Oliver Schaudt, and Maya Stein. A graph is H-free if it has no induced subgraph isomorphic to H. In the second part of this thesis, we characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable H-free graphs. This solves a problem posed by Golovach et al. We also characterize all graphs H for which there are only finitely many H-free minimal obstructions for list 3-colorability. This is joint work with Maria Chudnovsky, Jan Goedgebeur and Oliver Schaudt. The last result of this thesis deals with a scheduling problem addressing the uncertainty regarding the machines. We study a scheduling environment in which jobs first need to be grouped into some sets before the number of machines is known, and then the sets need to be scheduled on machines without being separated. In order to evaluate algorithms in such an environment, we introduce the idea of an alpha-robust algorithm, one which is guaranteed to return a schedule on any number m of machines that is within an alpha factor of the optimal schedule on m machines, where the optimum is not subject to the restriction that the sets cannot be separated. Under such environment, we give a (5/3+epsilon)-robust algorithm for scheduling on parallel machines to minimize makespan, and show a lower bound of 4/3. For the special case when the jobs are infinitesimal, we give a 1.233-robust algorithm with an asymptotic lower bound of 1.207. This is joint work with Clifford Stein.

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.

Researchers in science and engineering face various obstacles due to a lack of specific and full data. Many different approaches have been devised to deal with these restrictive requirements, but two notable schools of thought are the fuzzy set (FS) theory and the rough set (RS) theory, both of which have spawned many extensions and hybridizations. Although RS theory originated from an indiscernibility relation (also known as an equivalence relation in mathematics), emphasis rapidly shifted to similarity or coverings (and their fuzzy analogues). Many other hybrid schemes were suggested with this goal in mind. The gap between those concepts shrank because to this thorough analysis. Fuzzy set theory is a legitimate way to convey the ambiguity of assessment data, yet it is still inadequate for dealing with certain intricate problems in the actual world. In reality, decision makers will undoubtedly provide different kinds of ambiguous and nuanced assessments. Atanassov’s intuitionistic fuzzy set theory broadened the application of fuzzy set theory by imbuing it with an element of uncertainty. Sometimes in real life, you have to deal with a neutral element on top of the indeterminate one. Picture fuzzy sets were developed specifically for this purpose. Membership roles may be positive, neutral, or negative/refusal. In contrast, hesitant fuzzy sets and its hybrid models are useful when decision makers are on the fence about which option to choose. As a binary relation on a set, a graph is symmetric. It is a staple in mathematical modelling and is used in almost every scientific and technological discipline. Graph theory has been essential in the mathematical modelling and subsequent resolution of several real-world situations. Information about connections between things is often best represented using graph theory, which uses vertices to stand in for the items and edges for the relationships between them. The suggested dynamic algorithm is better to the static approach in dealing with the multidimensional dynamic changes of the hybrid incomplete decision system, according to a series of experiments carried out on nine UCI datasets.

A rough set method for the minimum vertex cover problem of graphs

We provide for the first time, a complete list of forbidden minors (obstructions) for the family of graphs with vertex cover 6. This study shows how to limit both the search space of graphs and improve the efficiency of an obstruction checking algorithm when restricted to k–VERTEX COVER graph families. In particular, our upper bounds 2k + 1 (2k + 2) on the maximum number of vertices for connected (disconnected) obstructions are shown to be sharp for all k > 0. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 163–178, 2002