求解最小Steiner树的蚁群优化算法及其收敛性

最小Steiner树问题是NP难问题,它在通信网络等许多实际问题中有着广泛的应用．蚁群优化算法是最近提出的求解复杂组合优化问题的启发式算法．本文以无线传感器网络中的核心问题之一,路由问题为例,给出了求解最小Steiner树的蚁群优化算法的框架．把算法的迭代过程看作是离散时间的马尔科夫过程,证明了在一定的条件下,该算法所产生的解能以任意接近于1的概率收敛到路由问题的最优解．     

时变挥发率条件下求解Steiner树蚁群优化算法的收敛性

蚁群优化算法是最近提出的求解复杂组合优化问题的启发式算法.在蚁群优化算法中,信息素的更新规则直接影响着算法性能,固定挥发率条件下,虽然也能得到求解Steinei树蚁群优化算法的收敛性结果,但算法的探优能力差,易于陷入局部最优.本文在设计求解最小Steiner树蚁群优化算法时,采用了动态更新信息索挥发率的方法,并给出了时变挥发率条件下算法的收敛性证明.具体的,在时变挥发率条件下,当迭代次数充分大时,该算法能以概率1找到最优解.另外,在动态更新信息素下界的条件下,也能得到类似的收敛性结果.     

遗传算法与蚂蚁算法的融合在带时间窗的车辆路径问题中的应用

对带时间窗的车辆路径问题(VRPTW)的求解分为两个过程,先由遗传算法求解出初步的可行解,由此生成信息素初始分布,而后采用蚂蚁算法找出问题的最优解或近似最优解.通过具体算例,从数值计算上探索了遗传算法和蚂蚁算法融合后的优化能力,获得了满意的效果.     

一种改进的混合型蚁群算法在TSP问题中的应用

介绍了一种求解TSP问题的算法改进的混合型蚁群算法,该算法在近邻法构造初始解的基础上,使用2-opt局部搜索法对当前解进行改进,在更新全局信息素时采用基于排序的蚂蚁系统对排在前2名的蚂蚁更新全局信息素,且为全局信息素设置最大值和最小值,并使用Matlab仿真求解了kroa200等13个经典tsp问题,得到的结果和最优解的误差很小,并和两种最新改进的蚁群算法以及两种自组织算法进行比较,比较结果充分证明了该改进算法的有效性.     

一种求解TSP问题的多策略改进蚁群算法

蚁群算法是一种求解复杂组合优化问题的启发式仿生进化算法,并是求解TSP问题行之有效的一种随机算法.但此算法仍存在求解精度低、易陷入局部最优及求解效率低的问题,针对该问题提出一种多策略改进蚁群算法.采用最近邻法影响初始信息素的分布,达到降低算法初期较短路径上信息素浓度的目的,并在转移规则变异调整的基础上,结合路径的均值交叉进化策略,增强算法探索全局解空间和避免陷入局部最优的能力.然后,结合迭代和精英策略对信息素更新机制进行改进,进一步提高化算法的求解性能及求解效率,最后,对从TSPLIB数据库选出的8个实例进行求解并与其他算法进行对比,实验结果表明,改进算法在求解旅行商问题时的高效性,且具有较高的运算性能.     

基于模拟退火算法的充电路径规划

主要利用模拟退火算法解决针对无线传感器网络的充电器路径规划问题,并求得网络中每个传感器对应的最小电池容量.该实际问题可抽象为经典旅行商问题(TSP)以及多旅行商问题(MTSP).针对中小规模的TSP问题,以总路程最小为优化目标,利用模拟退火算法搜索全局最优解;针对MTSP问题,以多条路径中最长的路程和每条支路平均路程的加权之和为优化目标,利用模拟退火算法进行求解.本文将最小电池容量模型简化为线性函数进行求解,并按照实际情况设计部分参数数值和部分参数取值范围,得到每个传感器最小电池容量的具体数值.     

一种基于边交换的贪心算法求解md-MST问题

通过对最小度限制最小生成树(md-MST)问题性质进行分析,提出了一种基于边交换的贪心算法.算法先用贪心算法生成一棵生成树ST,然后对生成树ST经过边交换调整,得到满足问题约束条件的可行解,再对生成树ST进行进一步边交换优化,得到md-MST问题的最优解或接近最优解的近似解.实验证明,算法能在短对间内求出大规模顶点随机图的md-MST,是一种非常实用的求解md-MST问题的精确算法.     

基于离散群居蜘蛛算法的WSNs分簇路由优化

对无线传感器网络(WSNs)路由优化问题进行研究,提出一种基于离散群居蜘蛛算法的WSNs分簇路由优化方案.首先定量分析节点覆盖冗余度期望值与网络覆盖率的关系,筛选出能够保证网络覆盖率要求的最少网络工作节点,其次研究分簇大小与网络节点密度的关系,动态地确定最佳的分簇个数.基于此,以簇间距离和簇首能量为评价指标构建簇间通信模型,重新定义蜘蛛个体编码方式和更新策略,采用离散群居蜘蛛算法对模型进行求解,最终实现WSNs分簇路由优化.仿真结果表明,方案能够满足网络覆盖要求,而且与其它路由优化算法相比,延长了网络生命周期,降低了网络能耗.     

求解多项式规划的一个全局最优化算法

提出一个求解带箱子约束的一般多项式规划问题的全局最优化算法, 该算法包含两个阶段, 在第一个阶段, 利用局部最优化算法找到一个局部最优解. 在第二阶段, 利用一个在单位球上致密的向量序列, 将多元多项式转化为一元多项式, 通过求解一元多项式的根, 找到一个比当前局部最优解更好的点作为初始点, 回到第一个 阶段, 从而得到一个更好的局部最优解, 通过两个阶段的循环最终找到问题的全局最优解, 并给出了算法收敛性分析. 最后, 数值结果表明了算法是有效的.     

基于信息素的改进蚁群算法及其在TSP中的应用

针对基本蚁群算法收敛速度慢、易陷于局部最优从而导致搜索停滞的缺陷,提出了一种改进蚁群算法模型.改进算法引入信息素调节系数,避免算法初期各路径上信息素出现过大差异,导致算法"早熟".通过动态调整信息素挥发,在求解速度和寻找全局最优之间寻找平衡.对旅行商问题的仿真结果表明:改进算法的求解结果和求解效率都明显优于基本蚁群算法.     

Steiner tree packing revisited

The Steiner tree packing problem (STPP) in graphs is a long studied problem in combinatorial optimization. In contrast to many other problems, where there have been tremendous advances in practical problem solving, STPP remains very difficult. Most heuristics schemes are ineffective and even finding feasible solutions is already NP-hard. What makes this problem special is that in order to reach the overall optimal solution non-optimal solutions to the underlying NP-hard Steiner tree problems must be used. Any non-global approach to the STPP is likely to fail. Integer programming is currently the best approach for computing optimal solutions. In this paper we review some ??classical?? STPP instances which model the underlying real world application only in a reduced form. Through improved modelling, including some new cutting planes, and by employing recent advances in solver technology we are for the first time able to solve those instances in the original 3D grid graphs to optimimality.     

Computing optimal rectilinear Steiner trees: A survey and experimental evaluation

The rectilinear Steiner tree problem is to find a minimum-length rectilinear interconnection of a set of points in the plane. A reduction from the rectilinear Steiner tree problem to the graph Steiner tree problem allows the use of exact algorithms for the graph Steiner tree problem to solve the rectilinear problem. Furthermore, a number of more direct, geometric algorithms have been devised for computing optimal rectilinear Steiner trees. This paper surveys algorithms for computing optimal rectilinear Steiner trees and presents experimental results comparing nine of them: graph Steiner tree algorithms due to Beasley, Bern, Dreyfus and Wagner, Hakimi, and Shore, Foulds, and Gibbons and geometric algorithms due to Ganley and Cohoon, Salowe and Warme, and Thomborson, Alpern, and Carter.     

Improving Construction for Connected Dominating Set with Steiner Tree in Wireless Sensor Networks

The connected dominating set plays an important role in ad hoc wireless networking. Many constructions for approximating the minimum connected dominating set have been proposed in the literature. In this paper, we propose a new one with Steiner tree, which produces approximation solution within a factor of 6.8 from optimal. This approximation algorithm can also be implemented distributedly.     

Local search for the Steiner tree problem in the Euclidean plane

Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative improvements using the minimum spanning tree as an initial solution. We may therefore characterize them as local search heuristics. In this paper, we first give a survey of existing heuristic approaches from a local search perspective, by setting up solution spaces and neighbourhood structures. Secondly, we present a new general local search approach which is based on a list of full Steiner trees constructed in a preprocessing phase. This list defines a solution space on which three neighbourhood structures are proposed and evaluated. Computational results show that this new approach is very competitive from a cost–benefit point of view. Furthermore, it has the advantage of being easy to apply to the Steiner tree problem in other metric spaces and to obstacle avoiding variants.     

A Subdivision Approach to Maximum Parsimony

Determining an optimal phylogenetic tree using maximum parsimony, also referred to as the Steiner tree problem in phylogenetics, is NP hard. Here we provide a new formulation for this problem which leads to an analytical and linear time solution when the dimensionality (sequence length, or number of characters) is at most two. This new formulation of the problem provides a direct link between the maximum parsimony problem and the maximum compatibility problem via the intersection graph. The solution for the “two character case” has numerous practical applications in phylogenetics, some of which are discussed. Received January 16, 2006     

三维欧氏Steiner最小树的Delaunay四面体网格混合智能算法

Steiner最小树问题是组合优化中经典的NP难题,在许多实际问题中有着广泛的应用,而三维欧氏Steiner最小树问题是对二维欧氏Steiner最小树问题的推广。由于三维欧氏Steiner树问题的求解非常困难,至今为止的相关成果较为少见。本文针对该问题,利用Delaunay四面体网格剖分技术,提出了一种混合型智能求解方法,不仅可以尽量避免拓扑结构陷入局部最优,且对较大规模的问题求解亦有良好的效果。算法在Matlab环境下编程实现,经实例测试,获得了满意的效果。     

Breakout local search for the Steiner tree problem with revenue,budget and hop constraints

The Steiner tree problem (STP) is one of the most popular combinatorial optimization problems with various practical applications. In this paper, we propose a Breakout Local Search (BLS) algorithm for an important generalization of the STP: the Steiner tree problem with revenue, budget and hop constraints (STPRBH), which consists of determining a subtree of a given undirected graph which maximizes the collected revenues, subject to both budget and hop constraints. Starting from a probabilistically constructed initial solution, BLS uses a Neighborhood Search (NS) procedure based on several specifically designed move operators for local optimization, and employs an adaptive diversification strategy to escape from local optima. The diversification mechanism is implemented by adaptive perturbations, guided by dedicated information of discovered high-quality solutions. Computational results based on 240 benchmarks show that BLS produces competitive results with respect to several previous approaches. For the 56 most challenging instances with unknown optimal results, BLS succeeds in improving 49 and matching one best known results within reasonable time. For the 184 instances which have been solved to optimality, BLS can also match 167 optimal results.     

A simple proof of Hwang's theorem for rectilinear Steiner minimal trees

We present a simple, direct proof of Hwang's characterization of rectilinear Steiner minimal trees [3]: LetS be a set of at least five terminals in the plane. If no rectilinear Steiner minimal tree forS has a terminal of degree two or more, there is a tree in which at most one of the Steiner points does not lie on a straight linel, and the tree edges incident to the Steiner points onl appear on alternate sides. This theorem has been found useful in proving other results for rectilinear Steiner minimal trees.     

An exact solution framework for the minimum cost dominating tree problem

The minimum cost dominating tree problem is a recently introduced NP-hard problem, which consists of finding a tree of minimal cost in a given graph, such that for every node of the graph, the node or one of its neighbours is in the tree. We present an exact solution framework combining a primal–dual heuristic with a branch-and-cut approach based on a transformation of the problem into a Steiner arborescence problem with an additional constraint. The effectiveness of our approach is evaluated on testbeds proposed in literature containing instances with up to 500 nodes. Our framework manages to solve all but four instances from literature to proven optimality within 3 h (most of them in a few seconds). We provide optimal solution values for 69 instances from literature for which the optimal solution was previously unknown.