The quadratic minimum spanning tree problem: A lower bounding procedure and an efficient search algorithm

In this paper we consider the quadratic minimum spanning tree problem (QMSTP) which is known to be NP-hard. Given a complete graph, the QMSTP consists of finding a minimum spanning tree (MST) where interaction costs between pairs of edges are prescribed. A Lagrangian relaxation procedure is devised and an efficient local search algorithm with tabu thresholding is developed. Computational experiments are reported on standard test instances, randomly generated test instances and quadratic assignment problem (QAP) instances from the QAPLIB by using a transformation scheme. The local search heuristic yields very good performance and the Lagrangian relaxation procedure gives the tightest lower bounds for all instances when compared to previous lower bounding approaches.     

An artificial bee colony algorithm for the minimum routing cost spanning tree problem

Given a connected, weighted, and undirected graph, the minimum routing cost spanning tree problem seeks a spanning tree of minimum routing cost on this graph, where routing cost of a spanning tree is defined as the sum of the costs of the paths connecting all possible pairs of distinct vertices in that spanning tree. This problem has several important applications in networks design and computational biology. In this paper, we have proposed an artificial bee colony (ABC) algorithm-based approach for this problem. We have compared our approach against four best methods reported in the literature—two genetic algorithms, a stochastic hill climber and a perturbation-based local search. Computational results show the superiority of our ABC approach over other approaches.     

Efficient minimum spanning tree construction without Delaunay triangulation

Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least Ω(n2) time. More efficient approaches find a minimum spanning tree only among edges in the Delaunay triangulation of the points. However, Delaunay triangulation is not well defined in rectilinear distance. In this paper, we first establish a framework for minimum spanning tree construction which is based on a general concept of spanning graphs. A spanning graph is a natural definition and not necessarily a Delaunay triangulation. Based on this framework, we then design an O(nlogn) sweep-line algorithm to construct a rectilinear minimum spanning tree without using Delaunay triangulation.     

Reconstructing a Minimum Spanning Tree after Deletion of Any Node

Abstract. Updating a minimum spanning tree (MST) is a basic problem for communication networks. In this paper we consider single node deletions in MSTs. Let G=(V,E) be an undirected graph with n nodes and m edges, and let T be the MST of G . For each node v in V , the node replacement for v is the minimum weight set of edges R(v) that connect the components of T-v . We present a sequential algorithm and a parallel algorithm that find R(v) for all V simultaneously. The sequential algorithm takes O(m log n) time, but only O(m α (m,n)) time when the edges of E are presorted by weight. The parallel algorithm takes O(log ² n) time using m processors on a CREW PRAM.     

Random-tree Diameter and the Diameter-constrained MST

A minimum spanning tree (MST) with a small diameter is required in numerous practical situations such as when distributed mutual-exclusion algorithms are used, or when information retrieval algorithms need to compromise between fast access and small storage. The Diameter-Constrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph, G , with n nodes and a positive integer, k , find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NP-complete, for all values of k ; 4 h k h ( n m 2). In this paper, we investigate the behavior of the diameter of an MST in randomly generated graphs. Then, we present heuristics that produce approximate solutions for the DCMST problem in polynomial time. We discuss convergence, relative merits, and implementation of these heuristics. Our extensive empirical study shows that the heuristics produce good solutions for a wide variety of inputs.     

The Stackelberg Minimum Spanning Tree Game

We consider a one-round two-player network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor??s prices). The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest possible minimum spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. This game is the minimum spanning tree analog of the well-studied Stackelberg shortest-path game. We analyze the complexity and approximability of the first player??s best strategy in StackMST. In particular, we prove that the problem is APX-hard even if there are only two different red costs, and give an approximation algorithm whose approximation ratio is at most min?{k,1+ln?b,1+ln?W}, where k is the number of distinct red costs, b is the number of blue edges, and W is the maximum ratio between red costs. We also give a natural integer linear programming formulation of the problem, and show that the integrality gap of the fractional relaxation asymptotically matches the approximation guarantee of our algorithm.     

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].     

交通选线优化算法的设计与实现

将交通选线问题求解转化为最小生成树（Minimun Spanning Tree,MST）的求解,对比了经典MST求解算法,以图论为基础,采取一种求最小生成树的改进遗传算法．该算法以二进制编码表示最小树问题,用深度优先搜索算法进行图的连通性判断,并采用相应的适应度函数、单亲换位算子和单亲逆转算子及多种控制进化策略,能在一次遗传进化过程中获得一批最小生成树,可供决策部门综合评价与决策．     

Spanning trees with minimum weighted degrees

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this problem within a 2−ε factor.     

Survivable network design: the capacitated minimum spanning network problem

We are given an undirected graph G=(V,E) with positive weights on its vertices representing demands, and non-negative costs on its edges. Also given are a capacity constraint k, and root vertex r∈V. In this paper, we consider the capacitated minimum spanning network (CMSN) problem, which asks for a minimum cost spanning network such that the removal of r and its incident edges breaks the network into a number of components (groups), each of which is 2-edge-connected with a total weight of at most k. We show that the CMSN problem is NP-hard, and present a 4-approximation algorithm for graphs satisfying triangle inequality. We also show how to obtain similar approximation results for a related 2-vertex-connected CMSN problem.     

A linear-time algorithm to compute a MAD tree of an interval graph

The average distance of a connected graph G is the average of the distances between all pairs of vertices of G. We present a linear time algorithm that determines, for a given interval graph G, a spanning tree of G with minimum average distance (MAD tree). Such a tree is sometimes referred to as a minimum routing cost spanning tree.     

Rough‐fuzzy quadratic minimum spanning tree problem

A quadratic minimum spanning tree problem determines a minimum spanning tree of a network whose edges are associated with linear and quadratic weights. Linear weights represent the edge costs whereas the quadratic weights are the interaction costs between a pair of edges of the graph. In this study, a bi‐objective rough‐fuzzy quadratic minimum spanning tree problem has been proposed for a connected graph, where the linear and the quadratic weights are represented as rough‐fuzzy variables. The proposed model is formulated by using rough‐fuzzy chance‐constrained programming technique. Subsequently, three related theorems are also proposed for the crisp transformation of the proposed model. The crisp equivalent models are solved with a classical multi‐objective solution technique, the epsilon‐constraint method and two multi‐objective evolutionary algorithms: (a) nondominated sorting genetic algorithm II (NSGA‐II) and (b) multi‐objective cross‐generational elitist selection, heterogeneous recombination, and cataclysmic mutation (MOCHC) algorithm. A numerical example is provided to illustrate the proposed model when solved with different methodologies. A sensitivity analysis of the example is also performed at different confidence levels. The performance of NSGA‐II and MOCHC are analysed on five randomly generated instances of the proposed model. Finally, a numerical illustration of an application of the proposed model is also presented in this study.     

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.     

A hybrid filter/wrapper approach of feature selection using information theory

We focus on a hybrid approach of feature selection. We begin our analysis with a filter model, exploiting the geometrical information contained in the minimum spanning tree (MST) built on the learning set. This model exploits a statistical test of relative certainty gain, used in a forward selection algorithm. In the second part of the paper, we show that the MST can be replaced by the 1 nearest-neighbor graph without challenging the statistical framework. This leads to a feature selection algorithm belonging to a new category of hybrid models (filter-wrapper). Experimental results on readily available synthetic and natural domains are presented and discussed.     

关于无穷计算问题的水平分裂算法分析及改进

首先介绍了用于路由选择的DVR(距离矢量路由)算法及其存在的无穷计算问题。然后阐述了用于解决该问题的水平分裂算法的思想,并运用MST(最小生成树)分析法对其进行了基于树型、环型和网状3种网络拓扑结构的算法分析。最后提出了一种改进的水平分裂算法———下一跳算法,并在实例分析的基础上对其存在的问题进行了总结。     

基于最小代价和生成树的算法研究

最小生成树问题是一类经典的组合优化问题,已有许多快速有效的算法。但是在实际中更多存在这样的网络,除边有权值外,结点也有权值,并且结点的权值有多种情况,这就产生了基于代价和最小的生成树问题。根据结点权值的取值情况,对几种最小代价和生成树问题进行分类和求解,得到了一些有价值的性质和算法,有一定的实际应用背景。     

Enhanced second order algorithm applied to the capacitated minimum spanning tree problem

Given a centralized undirected graph with costs associated with its edges, the capacitated minimum spanning tree problem is to find a minimum cost spanning tree of the given graph, subject to a capacity constraint in all subtrees incident in the central node. As the problem is NP-hard, we propose an enhanced version of the well-known second order algorithm, described in [Karnaugh M. A new class of algorithms for multipoint network optimization. IEEE Transactions on Communications 1976;COM-24:500–5.]. The original version of this algorithm is based on a look-ahead strategy, used for a tentative inclusion of a constraint to the problem, performed in each iteration. In the new enhanced version, we propose the inclusion of look-behind steps, which can be seen as the reverse of the look-ahead procedure. Therefore and using some memory features, the method can continue even when facing the traditional stopping criterion of the original algorithm. Computational experiments showing the effectiveness of the new method on benchmark instances are reported.     

Ant Colony Optimization and the minimum spanning tree problem

Ant Colony Optimization (ACO) is a kind of metaheuristic that has become very popular for solving problems from combinatorial optimization. Solutions for a given problem are constructed by a random walk on a so-called construction graph. This random walk can be influenced by heuristic information about the problem. In contrast to many successful applications, the theoretical foundation of this kind of metaheuristic is rather weak. Theoretical investigations with respect to the runtime behavior of ACO algorithms have been started only recently for the optimization of pseudo-Boolean functions.We present the first comprehensive rigorous analysis of a simple ACO algorithm for a combinatorial optimization problem. In our investigations, we consider the minimum spanning tree (MST) problem and examine the effect of two construction graphs with respect to the runtime behavior. The choice of the construction graph in an ACO algorithm seems to be crucial for the success of such an algorithm. First, we take the input graph itself as the construction graph and analyze the use of a construction procedure that is similar to Broder’s algorithm for choosing a spanning tree uniformly at random. After that, a more incremental construction procedure is analyzed. It turns out that this procedure is superior to the Broder-based algorithm and produces additionally in a constant number of iterations an MST, if the influence of the heuristic information is large enough.     

A one-parameter genetic algorithm for the minimum labeling spanning tree problem

Given a connected, undirected graph G whose edges are labeled (or colored), the minimum labeling spanning tree (MLST) problem seeks a spanning tree on G with the minimum number of distinct labels (or colors). In recent work, the MLST problem has been shown to be NP-hard and an effective heuristic [maximum vertex covering algorithm (MVCA)] has been proposed and analyzed. We use a one-parameter genetic algorithm (GA) to solve the problem. In computational tests, the GA clearly outperforms MVCA.     

Minimizing the makespan in a single machine scheduling problem with a time-based learning effect

Both the building cost and the multiple-source routing cost are important considerations in construction of a network system. A spanning tree with minimum building cost among all spanning trees is called a minimum spanning tree (MST), and a spanning tree with minimum k-source routing cost among all spanning trees is called a k-source minimum routing cost spanning tree (k-MRCT). This paper proposes an algorithm to construct a spanning tree T for a metric graph G with a source vertex set S such that the building cost of T is at most 1+2/(α−1) times of that of an MST of G, and the k-source routing cost of T is at most α(1+2(k−1)(n−2)/k(n+k−2)) times of that of a k-MRCT of G with respect to S, where α>1, k=|S| and n is the number of vertices of G.