求解度约束最小生成树的新的快速算法

针对度约束最小生成树问题,提出了一种新的快速算法。新的快速算法分为两个主要部分,第一部分从一棵最小生成树出发,构造一棵度约束树。第二部分设计了一种改进策略,从第一部分求得的度约束树出发,每次去掉树的一条边,将顶点按照连通性划分成两个集合,在不违反度约束的情况下,从这两个集合构成的边割中,选择一条权值减少最大的边添加到图中。通过大量的数值实验表明新的快速算法性能良好。     

求解度约束最小生成树的一种改进算法

度约束最小生成树问题是网络设计和优化中的一个NP-hard问题。提出一种求解网络G关于指定节点的最大度约束最小生成树的改进算法。算法在保证指定节点最大度的前提下,通过选取剩余边中权最小的边加入当前网络,得到网络G关于指定节点的最大度最小生成树,同时对算法的复杂度进行了分析。最后通过与其他算法的仿真比较,表明新算法的有效性和通用性。     

求解度约束最小生成树问题的新算法

针对网络设计和组合优化中的度约束最小生成树问题,基于第k最小生成树的求解算法,提出了一种求解网络G关于指定节点的最小k度生成树的新算法。该算法通过对网络G的最小生成树作最优可行变换,逐步构造出指定节点的度数越来越接近度约束k的最小i度生成树,最终得到了网络G关于指定节点的最小k度生成树。给出了算法实施的具体步骤,并证明了算法的正确性。最后通过仿真结果和一个运输实例,表明了该算法在解决度约束最小生成树问题中的有效性。     

An adaptive heuristic to the bounded-diameter minimum spanning tree problem

Given a graph G and a bound d?≥?2, the bounded-diameter minimum spanning tree problem seeks a spanning tree on G of minimum weight subject to the constraint that its diameter does not exceed d. This problem is NP-hard; several heuristics have been proposed to find near-optimal solutions to it in reasonable times. A decentralized learning automata-based algorithm creates spanning trees that honor the diameter constraint. The algorithm rewards a tree if it has the smallest weight found so far and penalizes it otherwise. As the algorithm proceeds, the choice probability of the tree converges to one; and the algorithm halts when this probability exceeds a predefined value. Experiments confirm the superiority of the algorithm over other heuristics in terms of both speed and solution quality.     

Solving diameter‐constrained minimum spanning tree problems by constraint programming

The diameter‐constrained minimum spanning tree problem consists in finding a minimum spanning tree of a given graph, subject to the constraint that the maximum number of edges between any two vertices in the tree is bounded from above by a given constant. This problem typically models network design applications where all vertices communicate with each other at a minimum cost, subject to a given quality requirement. We propose alternative formulations using constraint programming that circumvent weak lower bounds yielded by most mixed‐integer programming formulations. Computational results show that the proposed formulation, combined with an appropriate search procedure, solves larger instances and is faster than other approaches in the literature.     

求解度约束最小生成树的新的遗传算法

针对度约束最小生成树问题的特征,设计了一种新的编码方式,并在此基础上提出了一个新遗传算法来求解该问题。该算法采用新的启发式杂交算子、变异算子和局部搜索算子,以概率1收敛到全局最优解。数值实验表明该算法优于文中提出的其他4种算法。     

面向无线内容分发网络的树形拓扑生成算法

在无线内容分发网络中,为减轻骨干网络的传输压力,可将网络拓扑结构构建为以基站和Wi Fi接入点为根的若干棵最小生成树,并对生成树的深度和每个节点的度数进行约束。这种深度和度数约束的最小生成树问题是一个NP完全问题。针对该问题,首先提出能够生成优质近似解的启发式算法,该算法在不违反深度以及度数约束的情况下构建生成树,算法思想为在服务性节点相连的边中选择与当前生成树相连且权值最小的边加入生成树。然后在生成初始近似解的基础上采用定制的禁忌搜索算法和模拟退火算法对该近似解实施进一步优化。实验结果表明,在给定的约束条件下,禁忌搜索算法求得的解优于现有的遗传算法,在深度约束为4以及度数约束为10的条件下,解的改进幅度可达18.5%,所提算法的运行速度比遗传算法提高了10倍。     

Edge exchanges in the degree-constrained minimum spanning tree problem

We describe a branch and bound algorithm to solve the degree-constrained minimum spanning tree problem. We propose an edge exchange analysis frequently used in the algorithm and three types of heuristic methods. Computational results are reported for problems with up to 200 vertices. These results are much better than known results from the literature.     

A new evolutionary approach to the degree-constrained minimumspanning tree problem

Finding the degree-constrained minimum spanning tree (d-MST) of a graph is a well-studied NP-hard problem of importance in communications network design and other network-related problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a novel tree construction algorithm called the randomized primal method (RPM) which builds degree-constrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multistart hillclimbing, and a genetic algorithm. While other researchers have mainly concentrated on finding spanning trees in Euclidean graphs, we consider the more general case of random graph problems. We describe two random graph generators which produce particularly challenging d-MST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multistart hillclimbing. Our experimental results provide strong evidence that the genetic algorithm employing RPM finds significantly lower-cost solutions to random graph d-MST problems than rival methods     

Minimal spanning tree subject to a side constraint

This paper addresses the problem of finding a minimal spanning tree in a network subject to a knapsack-type constraint. The problem is shown to belong to the class of NP -complete problems. Certain properties of an optimal solution to this problem are established considering a bicriteria spanning tree problem. Based on these, an algorithm is proposed which is mainly a branch and bound scheme branching from the test upper bound and generating efficient frontiers successively. It also uses certain effective heuristics and bounds that are obtained in polynomially bounded operations. The algorithm is validated and a numerical example is included.     

一种融合模拟退火和单亲遗传的优化求解算法

度约束最小生成树是一个经典的组合优化NP难题,其在网络设计和优化中有广泛的应用;现有求解方法往往不能很好地兼顾求解效率和求解精度;为了在缩短求解时间的同时,更好地获得最优解,提出了一种结合模拟退火算法和单亲遗传算法的改进求解算法;首先,改进遗传算法中变异因子的生成方式,避免不可行解个体的产生,并且设计自适应变异率,以提高算法的求解效率;其次,针对单亲遗传算法仅有变异操作可能导致最优解个体跳跃的问题,结合模拟退火的思想,来保证解的全局最优性;最后,在具体的度约束最小生成树问题中进行了三组实验,从运行时间和最优解的情况等方面与传统单亲遗传算法进行对比,实验表明该算法在求解效率和获得最优解方面都有较好的改进效果。     

一种新的求解度约束最小生成树的遗传算法

染色体编码是遗传算法的关键内容,编码的优劣并直接影响算法的性能.提出了基于过程控制的生成树编码方法--PC编码.PC码为定长的整数向量,使用PC编码求解特定生成树问题时,首先选定的一个有效算法,并将修改为可控算法,然后用编码向量控制算法的运行过程,从面得到唯一生成树.为了求解度约束最小生成树(DCMST)问题,在D-Prim算法的基础上,设计r过程可控的度约束生成树构造PC-Prim算法.给出了以PC-Prim算法作为译码器的求解DC-MST问题的遗传算法.仿真结果表明遗传算法求解精度和运行时间均优于参与其他算法.     

Degree-constrained minimum spanning tree

In this paper the problem of a degree-constrained minimum spanning tree (DCMST) is defined. The problem is formulated as a linear 0–1 integer programming problem. A primal and a dual heuristic (construction) procedure and a branch-and-bound algorithm are proposed to construct a DCMST. These procedures are illustrated with a simple example. Some computational experience with these algorithms is also reported.     

传输子网选择:度数有界最大支撑子图逼近

研究了源于无线网状网络的度数有界最大支撑子图问题:给定连通图G=(V,E)和正整数d≥2,求G的一个最大支撑子图H,满足对V中每个顶点v,v在H中的度数dH(v)不超过d。这里,支撑子图指图G的一个连通而且包括G中所有顶点的子图。就输入图的边是否带权,分别设计了多项式时间近似算法。当输入图为无权图时,证明了近似算法的近似比为2;当输入图为赋权图时,证明了算法输出一个最大度数不超过d+1、权重不低于最优解权重1/(d+2)的支撑子图。算法输出的度数有界支撑子图可以用作无线网状网络的传输子网。     

基于“最小差值法”的网络模型耗费代价研究

最小生成树算法是数据结构中,求网络模型耗费代价最优解的一个重要工具。现实生活中的连通网络模型复杂而多变,有时还需兼顾其它的目标,一棵最小生成树不足以解决问题,因此找出所有的最小生成树是很有必要的,在此提出一种新的寻找所有最小生成树的算法--最小差值法。无向连通图网络通过去掉连枝生成最小生成树,一个连枝加入最小生成树形成一个圈。这种算法是在一个圈中,用连枝的权与其它树枝的权分别作差,求最小差值。由最小差值是否为零,判断原有的最小生成树能否通过换进换出边,生成新的最小生成树。该算法能够有规律、高效率的寻找出所有的最小生成树。在找出的所有最小生成树方案中,选择符合实时情况的最小生成树方案,该方案即为网络耗费代价的最优解。     

Variable neighborhood search for the cost constrained minimum label spanning tree and label constrained minimum spanning tree problems

Given an undirected graph whose edges are labeled or colored, edge weights indicating the cost of an edge, and a positive budget B, the goal of the cost constrained minimum label spanning tree (CCMLST) problem is to find a spanning tree that uses the minimum number of labels while ensuring its cost does not exceed B. The label constrained minimum spanning tree (LCMST) problem is closely related to the CCMLST problem. Here, we are given a threshold K on the number of labels. The goal is to find a minimum weight spanning tree that uses at most K distinct labels. Both of these problems are motivated from the design of telecommunication networks and are known to be NP-complete [15].In this paper, we present a variable neighborhood search (VNS) algorithm for the CCMLST problem. The VNS algorithm uses neighborhoods defined on the labels. We also adapt the VNS algorithm to the LCMST problem. We then test the VNS algorithm on existing data sets as well as a large-scale dataset based on TSPLIB [12] instances ranging in size from 500 to 1000 nodes. For the LCMST problem, we compare the VNS procedure to a genetic algorithm (GA) and two local search procedures suggested in [15]. For the CCMLST problem, the procedures suggested in [15] can be applied by means of a binary search procedure. Consequently, we compared our VNS algorithm to the GA and two local search procedures suggested in [15]. The overall results demonstrate that the proposed VNS algorithm is of high quality and computes solutions rapidly. On our test datasets, it obtains the optimal solution in all instances for which the optimal solution is known. Further, it significantly outperforms the GA and two local search procedures described in [15].     

A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs

During the last decades, a host of efficient algorithms have been developed for solving the minimum spanning tree problem in deterministic graphs, where the weight associated with the graph edges is assumed to be fixed. Though it is clear that the edge weight varies with time in realistic applications and such an assumption is wrong, finding the minimum spanning tree of a stochastic graph has not received the attention it merits. This is due to the fact that the minimum spanning tree problem becomes incredibly hard to solve when the edge weight is assumed to be a random variable. This becomes more difficult if we assume that the probability distribution function of the edge weight is unknown. In this paper, we propose a learning automata-based heuristic algorithm to solve the minimum spanning tree problem in stochastic graphs wherein the probability distribution function of the edge weight is unknown. The proposed algorithm taking advantage of learning automata determines the edges that must be sampled at each stage. As the presented algorithm proceeds, the sampling process is concentrated on the edges that constitute the spanning tree with the minimum expected weight. The proposed learning automata-based sampling method decreases the number of samples that need to be taken from the graph by reducing the rate of unnecessary samples. Experimental results show the superiority of the proposed algorithm over the well-known existing methods both in terms of the number of samples and the running time of algorithm.     

度约束QoS组播路由遗传算法

有度约束的QoS组播路由问题在通信网络中具有重要意义。提出一种基于遗传算法的度约束组播路由算法,采用节点连接路径形式的编码方法构成一棵组播树的表示,设计了相应的具有树形结构的交叉和变异算子,以及节点度的改变算法。算法可以实现具有树形结构染色体的遗传进化。数值实验表明算法具有找到最优解的能力,特别适合于求解大规模网络有度约束的QoS组播路由问题。     

求解多目标最小生成树的改进多目标蚁群算法

多目标最小生成树问题是典型的NP问题。针对此问题,提出一种改进的多目标蚁群算法。为获得更好的非劣前端,通过合理选取多个信息素扩散源与扩散策略来避免其早熟收敛,并引入非支配排序算子,提高种群多样性并避免算法过早陷入局部最优解。对比实验结果表明：对于多目标最小生成树问题,该算法是有效的,不但在求解效率和解的质量方面优于相关算法,而且随着问题规模的扩大,算法仍保持较好的性能。     

An Explicit Solution of a Generalized Optimum Requirement Spanning Tree Problem With a Property Related to Monge

The paper considers a generalization of the optimum requirement spanning tree problem (ORST problem) first studied by Hu in 1974. Originally, ORST was regarded as a communication network of tree type with the minimum average cost, and it is obtained by the well-known Gomory–Hu algorithm when the degrees of vertices are not restricted. The ORST problem is generalized by (i) generalizing the objective function and (ii) imposing maximum degree constraints. The generalized ORST problem includes some practical problems, one of which is proposed in this paper, but is not efficiently solvable in general. However, I show that a particular tree (which is obtained by a sort of greedy algorithm but is explicitly definable) is a solution of the generalized problem when a certain practical condition is satisfied. The condition is closely related to the Monge property, which is originally discussed in the Hitchcock transportation problem, and is known to make some NP-hard problems efficiently solvable.