4类图完美匹配数目的显式表达式

匹配计数理论是图论的核心内容之一,此问题有很强的物理学、计算机科学和化学背景;但是,一般图的完美匹配计数问题却是[NP-]难问题。用划分、求和、再嵌套递推的方法给出了4类图完美匹配数目的显式表达式;所给出的方法,可以计算出相同结构重复出现的许多图的所有完美匹配的数目。     

A structure theorem for maximum internal matchings in graphs

A counterpart of the Gallai-Edmonds Structure Theorem is proved for matchings that are not required to cover the external vertices of graphs. The appropriate counterpart of Tutte's Theorem is derived from this result for the existence of perfect internal matchings in graphs.     

Conditional matching preclusion for the arrangement graphs

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper we find this number and classify all optimal sets for the arrangement graphs, one of the most popular interconnection networks.     

Partition-distance: A problem and class of perfect graphs arising in clustering

Partitioning of a set of elements into disjoint clusters is a fundamental problem that arises in many applications. Different methods produce different partitions, so it is useful to have a measure of the similarity, or distance, between two or more partitions. In this paper we examine one distance measure used in a clustering application in computational genetics. We show how to efficiently compute the distance, and how this defines a new class of perfect graphs.     

On the stability number of the edge intersection of two graphs

Let G be the graph obtained as the edge intersection of two graphs G₁, G₂ on the same vertex set V. We show that if at least one of G₁, G₂ is perfect, then α(G)?α(G₁)·α(G₂), where α() is the cardinality of the largest stable set. Moreover, for general G₁ and G₂, we show that α(G)?R(α(G₁)+1,α(G₂)+1)−1, where R(k,?) is the Ramsey number.     

A bound on the number of integrators needed to linearize a control system

This paper considers the problem of dynamic feedback linearization for nonlinear control systems. For a restricted class of dynamic compensators that correspond to adding chains of integrators to the inputs, we give an upper bound on the order of the compensator that needs to be considered. In the case of 2-input systems, we show that this bound is sharp.     

A General Class of Heuristics for Minimum Weight Perfect Matching and Fast Special Cases with Doubly and Triply Logarithmic Errors

We give a class of heuristic algorithms for minimum weight perfect matching on a complete edge-weighted graph K(V) satisfying the triangle inequality, where V is a set of an even number, n, of vertices. This class is a generalization of the Onethird heuristics, the hypergreedy heuristic, and it possibly employs any given exact or approximate perfect matching algorithm as an auxiliary heuristic to an appropriate subgraph of K(V). In particular, by using the heuristic of Gabow et al. [3] as its auxiliary heuristic, our algorithm can obtain a solution whose weight is at most times the weight of the optimal solution in time, or a solution with an error of in O(n ² ) time. Received July 21, 1994; revised November 28, 1995.     

A modified genetic algorithm approach for scheduling of perfect maintenance in distributed production scheduling

Distributed Scheduling (DS) problems have attracted attention by researchers in recent years. DS problems in multi-factory production are much more complicated than classical scheduling problems because they involve not only the scheduling problems in a single factory, but also the problems in the higher level, which is: how to allocate the jobs to suitable factories. It mainly focuses on solving two issues simultaneously: (i) allocation of jobs to suitable factories and (ii) determination of the corresponding production schedules in each factory. Its objective is to maximize system efficiency by finding an optimal plan for a better collaboration among various processes. However, in many papers, machine maintenance has usually been ignored during the production scheduling. In reality, every machine requires maintenance, which will directly influence the machine's availability, and consequently the planned production schedule. The objective of this paper is to propose a modified genetic algorithm approach to deal with those DS models with maintenance consideration, aiming to minimize the makespan of the jobs. Its optimization performance has been compared with other existing approaches to demonstrate its reliability. This paper also tests the influence of the relationship between the maintenance repairing time and the machine age to the performance of scheduling of maintenance during DS in the studied models.     

A limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G_n} be a sequence of graphs G₀,G₁,G₂,… that belong to a particular class. We consider graphs of the form K_n−G_n that result from the complete graph K_n after removing a set of edges that span G_n. We study the spanning tree behaviour of the sequence {K_n−G_n} when n→∞ and the number of edges of G_n scales according to n. More specifically, we define the spanning tree indicator ({G_n}), a quantity that characterizes the spanning tree behaviour of {K_n−G_n}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).     

A note on the Hadwiger number of circular arc graphs

The intention of this note is to motivate the researchers to study Hadwiger's conjecture for circular arc graphs. Let η(G) denote the largest clique minor of a graph G, and let χ(G) denote its chromatic number. Hadwiger's conjecture states that η(G)?χ(G) and is one of the most important and difficult open problems in graph theory. From the point of view of researchers who are sceptical of the validity of the conjecture, it is interesting to study the conjecture for graph classes where η(G) is guaranteed not to grow too fast with respect to χ(G), since such classes of graphs are indeed a reasonable place to look for possible counterexamples. We show that in any circular arc graph G, η(G)?2χ(G)−1, and there is a family with equality. So, it makes sense to study Hadwiger's conjecture for this family.     

Bounds for the 2-domination number of toroidal grid graphs

A k-dominating set for a graph G(V, E) is a set of vertices D? V such that every vertex v∈V\ D is adjacent to at least k vertices in D. The k-domination number of G, denoted by γ _k (G), is the cardinality of a smallest k-dominating set of G. Here we establish lower and upper bounds of γ _k (C _m ×C _n ) for k=2. In some cases, these bounds agree so that the exact 2-domination number is obtained.     

A long trip in the charming world of graphs for Pattern Recognition

This paper is a historical overview of graph-based methodologies in Pattern Recognition in the last 40 years; history is interpreted with the aim of recognizing the rationale inspiring the papers published in these years, so as to roughly classify them. Despite the extent of scientific production in this field, it is possible to identify three historical periods, each having its own connotation common to most of the corresponding papers, which are called here as the pure, the impure and extreme periods.     

A fast heuristic based on spacefilling curves for minimum-weight matching in the plane

We present a heuristic to find a good matching on n points in the plane. It is essentially sorting and so runs in O(n log n) worst-case time and linear expected time. Its performance is competitive with that of previously suggested methods. However, it has the advantages of being trivial to code and of being indifferent to the choice of metric or the probability distribution from which the points are drawn.     

A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints

The minimization problem of a quadratic objective function with the max-product fuzzy relation inequality constraints is studied in this paper. In this problem, its objective function is not necessarily convex. Hence, its Hessian matrix is not necessarily positive semi-definite. Therefore, we cannot apply the modified simplex method to solve this problem, in a general case. In this paper, we firstly study the structure of its feasible domain. We then use some properties of n × n real symmetric indefinite matrices, Cholesky’s decomposition, and the least square technique, and convert the problem to a separable programming problem. Furthermore, a relation in terms of a closed form is presented to solve it. Finally, an algorithm is proposed to solve the original problem. An application example in the economic area is given to illustrate the problem. Of course, there are other application examples in the area of digital data service and reliability engineering.     

A quadratic time algorithm to maximize the number of just-in-time jobs on identical parallel machines

In this paper, we study a scheduling problem on identical parallel machines, in which a processing time and a due date are given for each job, and the objective is to maximize the number of just-in-time jobs. A job is called just-in-time if it is completed exactly on its due date. We discuss the known results, show that a recently published greedy algorithm for this problem is incorrect, and present a new quadratic time algorithm which solves the problem.     

A graphical test for the interval stability of fractional-delay systems

This paper investigates the BIBO (bounded input and bounded output) interval stability testing of fractional-delay systems, a problem of justifying the BIBO stability of a polytopic family of functions involving fractional-order powers as well as exponential powers. It proves that the BIBO stability of the polytope is governed by the BIBO stability of the edges of the polytope, and the latter can be tested graphically via frequency response plots. The main results generalize some results in the literature.     

Determination of the optimal number of features for quadratic discriminant analysis via the normal approximation to the discriminant distribution

Given the joint feature-label distribution, increasing the number of features always results in decreased classification error; however, this is not the case when a classifier is designed via a classification rule from sample data. Typically, for fixed sample size, the error of a designed classifier decreases and then increases as the number of features grows. The problem is especially acute when sample sizes are very small and the potential number of features is very large. To obtain a general understanding of the kinds of feature-set sizes that provide good performance for a particular classification rule, performance must be evaluated based on accurate error estimation, and hence a model-based setting for optimizing the number of features is needed. This paper treats quadratic discriminant analysis (QDA) in the case of unequal covariance matrices. For two normal class-conditional distributions, the QDA classifier is determined according to a discriminant. The standard plug-in rule estimates the discriminant from a feature-label sample to obtain an estimate of the discriminant by replacing the means and covariance matrices by their respective sample means and sample covariance matrices. The unbiasedness of these estimators assures good estimation for large samples, but not for small samples.Our goal is to find an essentially analytic method to produce an error curve as a function of the number of features so that the curve can be minimized to determine an optimal number of features. We use a normal approximation to the distribution of the estimated discriminant. Since the mean and variance of the estimated discriminant will be exact, these provide insight into how the covariance matrices affect the optimal number of features. We derive the mean and variance of the estimated discriminant and compare feature-size optimization using the normal approximation to the estimated discriminant with optimization obtained by simulating the true distribution of the estimated discriminant. Optimization via the normal approximation to the estimated discriminant provides huge computational savings in comparison to optimization via simulation of the true distribution. Feature-size optimization via the normal approximation is very accurate when the covariance matrices differ modestly. The optimal number of features based on the normal approximation will exceed the actual optimal number when there is large disagreement between the covariance matrices; however, this difference is not important because the true misclassification error using the number of features obtained from the normal approximation and the number obtained from the true distribution differ only slightly, even for significantly different covariance matrices.