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 swarm intelligence approach to the quadratic minimum spanning tree problem

The quadratic minimum spanning tree problem (Q-MST) is an extension of the minimum spanning tree problem (MST). In Q-MST, in addition to edge costs, costs are also associated with ordered pairs of distinct edges and one has to find a spanning tree that minimizes the sumtotal of the costs of individual edges present in the spanning tree and the costs of the ordered pairs containing only edges present in the spanning tree. Though MST can be solved in polynomial time, Q-MST is NP-Hard. In this paper we present an artificial bee colony (ABC) algorithm to solve Q-MST. The ABC algorithm is a new swarm intelligence approach inspired by intelligent foraging behavior of honey bees. Computational results show the effectiveness of our approach.     

The multi-criteria minimum spanning tree problem based genetic algorithm

Minimum spanning tree (MST) problem is of high importance in network optimization and can be solved efficiently. The multi-criteria MST (mc-MST) is a more realistic representation of the practical problems in the real world, but it is difficult for traditional optimization technique to deal with. In this paper, a non-generational genetic algorithm (GA) for mc-MST is proposed. To keep the population diversity, this paper designs an efficient crossover operator by using dislocation a crossover technique and builds a niche evolution procedure, where a better offspring does not replace the whole or most individuals but replaces the worse ones of the current population. To evaluate the non-generational GA, the solution sets generated by it are compared with solution sets from an improved algorithm for enumerating all Pareto optimal spanning trees. The improved enumeration algorithm is proved to find all Pareto optimal solutions and experimental results show that the non-generational GA is efficient.     

A filter-and-fan algorithm for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) is a notoriously difficult problem in combinatorial optimization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new CMST heuristic algorithm that effectively combines the classical node-based and tree-based neighborhoods embodied in a filter-and-fan (F&F) approach, a local search procedure that generates compound moves in a tree search fashion. The overall algorithm is guided by a multi-level oscillation strategy used to trigger each type of neighborhood while allowing the search to cross feasibility boundaries. Computational results carried out on a standard set of 135 benchmark problems show that a simple F&F design competes effectively with prior CMST metaheuristics, rivaling the best methods, which are significantly more complex.     

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。     

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.     

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

We address the quadratic minimum spanning tree problem (QMSTP), the problem of finding a spanning tree of a connected and undirected graph such that a quadratic cost function is minimized. We first propose an integer programming formulation based on the reformulation–linearization technique (RLT). We then use the idea of partitioning spanning trees into forests of a given fixed size and obtain a QMSTP reformulation that generalizes the RLT model. The reformulation is such that the larger the size of the forests, the stronger lower bounds provided. Thus, a hierarchy of formulations is obtained. At the lowest hierarchy level, one has precisely the RLT formulation, which is already stronger than previous formulations in the literature. The highest hierarchy level provides the convex hull of integer feasible solutions for the problem. The formulations introduced here are not compact, so the direct evaluation of their linear programming relaxation bounds is not practical. To overcome that, we introduce two lower bounding procedures based on Lagrangian relaxation. These procedures are embedded into two parallel branch-and-bound algorithms. As a result of our study, several instances in the literature were solved to optimality for the first time.     

A simpler minimum spanning tree verification algorithm

The problem considered here is that of determining whether a given spanning tree is a minimal spanning tree. In 1984 Komlós presented an algorithm which required only a linear number of comparisons, but nonlinear overhead to determine which comparisons to make. We simplify his algorithm and give a linear-time procedure for its implementation in the unit cost RAM model. The procedure uses table lookup of a few simple functions, which we precompute in time linear in the size of the tree.     

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。     

A biased random-key genetic algorithm for the capacitated minimum spanning tree problem

This paper focuses on the capacitated minimum spanning tree (CMST) problem. Given a central processor and a set of remote terminals with specified demands for traffic that must flow between the central processor and terminals, the goal is to design a minimum cost network to carry this demand. Potential links exist between any pair of terminals and between the central processor and the terminals. Each potential link can be included in the design at a given cost. The CMST problem is to design a minimum-cost network connecting the terminals with the central processor so that the flow on any arc of the network is at most Q. A biased random-key genetic algorithm (BRKGA) is a metaheuristic for combinatorial optimization which evolves a population of random vectors that encode solutions to the combinatorial optimization problem. This paper explores several solution encodings as well as different strategies for some steps of the algorithm and finally proposes a BRKGA heuristic for the CMST problem. Computational experiments are presented showing the effectiveness of the approach: Seven new best-known solutions are presented for the set of benchmark instances used in the experiments.     

A dandelion-encoded evolutionary algorithm for the delay-constrained capacitated minimum spanning tree problem

This paper proposes an evolutionary algorithm with Dandelion-encoding to tackle the Delay-Constrained Capacitated Minimum Spanning Tree (DC-CMST) problem. This problem has been recently proposed, and consists of finding several broadcast trees from a source node, jointly considering traffic and delay constraints in trees. A version of the problem in which the source node is also included in the optimization process is considered as well in the paper. The Dandelion code used in the proposed evolutionary algorithm has been recently proposed as an effective way of encoding trees in evolutionary algorithms. Good properties of locality has been reported on this encoding, which makes it very effective to solve problems in which the solutions can be expressed in form of trees. In the paper we describe the main characteristics of the algorithm, the implementation of the Dandelion-encoding to tackled the DC-CMST problem and a modification needed to include the source node in the optimization. In the experimental section of this article we compare the results obtained by our evolutionary with that of a recently proposed heuristic for the DC-CMST, the Least Cost (LC) algorithm. We show that our Dandelion-encoded evolutionary algorithm is able to obtain better results that the LC in all the instances tackled.     

Computing all efficient solutions of the biobjective minimum spanning tree problem

A common way of computing all efficient (Pareto optimal) solutions for a biobjective combinatorial optimisation problem is to compute first the extreme efficient solutions and then the remaining, non-extreme solutions. The second phase, the computation of non-extreme solutions, can be based on a “k-best” algorithm for the single-objective version of the problem or on the branch-and-bound method. A k-best algorithm computes the k-best solutions in order of their objective values. We compare the performance of these two approaches applied to the biobjective minimum spanning tree problem. Our extensive computational experiments indicate the overwhelming superiority of the k-best approach. We propose heuristic enhancements to this approach which further improve its performance.     

The capacitated minimum spanning tree problem: revisiting hop-indexed formulations

The capacitated minimum spanning tree problem is to find a minimum spanning tree with an additional cardinality constraint on the number of nodes in any subtree off a given root node. In this paper we propose two improvements on a previous cutting-plane method proposed by Gouveia and Martins (Networks 35(1) (2000) 1) namely, a new set of inequalities that can be seen as hop-indexed generalization of the well known generalized subtour elimination (GSE) constraints and an improved separation heuristic for the original set of GSE constraints. Computational results show that the inclusion of the new separation routine and the inclusion of the new inequalities in Gouveia and Martins’ iterative method (see (Networks 35(1) (2000) 1)) produce improvements on previously reported lower bounds. Furthermore, with the improved method, several of previous unsolved instances have been solved to optimality.     

Randomized local search,evolutionary algorithms,and the minimum spanning tree problem

Randomized search heuristics, among them randomized local search and evolutionary algorithms, are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization. The analysis of these randomized search heuristics has been started for some well-known problems, and this approach is followed here for the minimum spanning tree problem. After motivating this line of research, it is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms.     

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].     

最小耗费生成树剔除算法及其正确性证明

文章提出了一种新的最小耗费生成树的算法，并对其正确性进行了证明。该算法通过从原图中逐步别除边来形成生成树，特别适用于当原图中边数较少(相对于顶点数)，或原图规模不大的情形。