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.     

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.     

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.     

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。     

Design of capacitated minimum spanning tree with uncertain cost and demand parameters

The classical Capacitated Minimum Spanning Tree Problem (CMSTP) deals with finding a minimum-cost spanning tree so that the total demand of the vertices in each subtree does not exceed the capacity limitation. In most of the CMSTP models, the edge costs and the demands of the vertices in the network are assumed to be known with certainty. This paper considers the CMSTP model, where the edge costs and/or the demands are only approximately known. A fast approximate reasoning algorithm, which is based on the Esau-Williams savings heuristic and fuzzy logic rules, is proposed. The computational results of the study based on the proposed approach are also reported.     

Ant-Tree: an ant colony optimization approach to the generalized minimum spanning tree problem

The ant colony optimization is a meta-heuristic inspired by knowledge sharing amongst ants using pheromone, which serves as a kind of collective memory. Since the past few years, there have been several successful applications of this new approach for finding approximate solutions for computationally difficult problems in reasonable times. In this paper, we study the generalized minimum spanning tree problem that involves the design of a minimum weight connected network spanning at least one node out of every disjoint subset of the nodes in a graph. This problem has a wealth of pertinence to a wide range of applications in different areas. As the problem is known as computationally challenging, we adopt the ant colony optimization strategy and present a new solution method, called Ant-Tree, to develop approximate solutions. As an initial attempt, our study aims to provide an investigation of the ant colony optimization approach for coping with tree optimization problems. Through computational experiments, we compare the performances of our approach and the method available in the literature. Numerical results indicate that the proposed method is effective in producing quality approximate solutions.     

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.     

Improved heuristics for the bounded-diameter minimum spanning tree problem

Given an undirected, connected, weighted graph and a positive integer k, the bounded-diameter minimum spanning tree (BDMST) problem seeks a spanning tree of the graph with smallest weight, among all spanning trees of the graph, which contain no path with more than k edges. In general, this problem is NP-Hard for 4 ≤ k < n − 1, where n is the number of vertices in the graph. This work is an improvement over two existing greedy heuristics, called randomized greedy heuristic (RGH) and centre-based tree construction heuristic (CBTC), and a permutation-coded evolutionary algorithm for the BDMST problem. We have proposed two improvements in RGH/CBTC. The first improvement iteratively tries to modify the bounded-diameter spanning tree obtained by RGH/CBTC so as to reduce its cost, whereas the second improves the speed. We have modified the crossover and mutation operators and the decoder used in permutation-coded evolutionary algorithm so as to improve its performance. Computational results show the effectiveness of our approaches. Our approaches obtained better quality solutions in a much shorter time on all test problem instances considered.     

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。     

最小比率生成树的竞争决策算法

最小比率生成树是找出目标函数形式为两个线性函数比值最小的生成树,例如总代价与总收益比值最小的生成树。当不限制分母的符号时,这是一个NP-hard问题。在分析最小比率生成树数学性质的基础上,提出了最小比率生成树的竞争决策算法。为了防止算法陷入局部最优,采用edge_exchange操作来增加算法的搜索范围。为了验证算法的有效性,采用无关和相关两种策略产生测试数据,并使用Delphi 7.0实现了算法的具体步骤。     

Degree-constrained minimum spanning tree problem with uncertain edge weights

Uncertainty theory has shown great advantages in solving many nondeterministic problems, one of which is the degree-constrained minimum spanning tree (DCMST) problem in uncertain networks. Based on different criteria for ranking uncertain variables, three types of DCMST models are proposed here: uncertain expected value DCMST model, uncertain α-DCMST model and uncertain most chance DCMST model. In this paper, we give their uncertainty distributions and fully characterize uncertain expected value DCMST and uncertain α-DCMST in uncertain networks. We also discover an equivalence relation between the uncertain α-DCMST of an uncertain network and the DCMST of the corresponding deterministic network. Finally, a related genetic algorithm is proposed here to solve the three models, and some numerical examples are provided to illustrate its effectiveness.     

An ant colony algorithm for the multi-compartment vehicle routing problem

We demonstrate the use of Ant Colony System (ACS) to solve the capacitated vehicle routing problem associated with collection of recycling waste from households, treated as nodes in a spatial network. For networks where the nodes are concentrated in separate clusters, the use of k-means clustering can greatly improve the efficiency of the solution. The ACS algorithm is extended to model the use of multi-compartment vehicles with kerbside sorting of waste into separate compartments for glass, paper, etc. The algorithm produces high-quality solutions for two-compartment test problems.     

An algorithm based on ant colony optimization for the minimum connected dominating set problem

Ant colony optimization is a well established metaheuristic from the swarm intelligence field for solving difficult optimization problems. In this work we present an application of ant colony optimization to the minimum connected dominating set problem, which is an NP-hard combinatorial optimization problem. Given an input graph, valid solutions are connected subgraphs of the given input graph. Due to the involved connectivity constraints, out-of-the-box integer linear programming solvers do not perform well for this problem. The developed ant colony optimization algorithm uses reduced variable neighborhood search as a sub-routine. Moreover, it can be applied to the weighted and to the non-weighted problem variants. An extensive experimental evaluation presents the comparison of our algorithm with the respective state-of-the-art techniques from the literature. It is shown that the proposed algorithm outperforms the current state of the art for both problem variants. For comparison purposes we also develop a constraint programming approach based on graph variables. Even though its performance deteriorates with growing instance size, it performs surprisingly well, solving 315 out of 481 considered problem instances to optimality.     

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。     

Nodal ant colony optimization for solving profit based unit commitment problem for GENCOs

This paper proposes a nodal ant colony optimization (NACO) technique to solve profit based unit commitment problem (PBUCP). Generation companies (GENCOs) in a competitive restructured power market, schedule their generators with an objective to maximize their own profit without any regard for system social benefit. Power and reserve prices become important factors in decision process. Ant colony optimization that mimics the behavior of ants foraging activities is suitably implemented to search the UCP search space. Here a search space consisting of optimal combination of binary nodes for unit ON/OFF status is represented for the movement of the ants to maintain good exploration and exploitation search capabilities. The proposed model help GENCOs to make decisions on the quantity of power and reserve that must be put up for sale in the markets and also to schedule generators in order to receive the maximum profit. The effectiveness of the proposed technique for PBUCP is validated on 10 and 36 generating unit systems available in the literature. NACO yields an increase of profit, greater than 1.5%, in comparison with the basic ACO, Muller method and hybrid LR-GA.     

基于最小生成树的图像分割

基于最小生成树思想,给出了一种利用改进的最小生成树进行图像分割的方案,减少了最小生成树的构建过程,对初分割的结果利用NNG算法进行合并。该方案节约了分割时间,并且对分割后的图像进行了有效的合并,达到了较好的分割效果。     

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。     

A 3.4713-approximation algorithm for the capacitated multicast tree routing problem

Given an underlying communication network represented as an edge-weighted graph G=(V,E), a source node sV, a set of destination nodes DV, and a capacity k which is a positive integer, the capacitated multicast tree routing problem asks for a minimum cost routing scheme for source s to send data to all destination nodes, under the constraint that in each routing tree at most k destination nodes are allowed to receive the data copies. The cost of the routing scheme is the sum of the costs of all individual routing trees therein. Improving on our previous approximation algorithm for the problem, we present a new algorithm which achieves a worst case performance ratio of , where ρ denotes the best known approximation ratio for the Steiner minimum tree problem. Since ρ is about 1.55 at the writing of the paper, the ratio achieved by our new algorithm is less than 3.4713. In comparison, the previously best ratio was .     

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.     

改进的最小生成树自适应分层聚类算法

针对传统最小生成树聚类算法需要事先知道聚类数目和使用静态全局分类依据,导致聚类密度相差较大时,算法有效性下降,计算复杂度大等问题,提出一种改进的最小生成树自适应分层聚类算法,根据最近邻关系,自动为每个聚类簇设定独立的阈值,使之适应分布密度相差较大的情况,并能自动确定聚类数目。实验表明,算法具有较好的性能,尤其对数据密度分布不均匀的情况也能得到较好的聚类结果。