Two new criteria for finding Steiner hulls in Steiner tree problems

The Steiner tree problem considered in this paper is that of finding a network of minimum length connecting a given setK of terminals in a regionR of the Euclidean plane. ASteiner hull forK inR is any subregion ofR known to contain a Steiner tree forK inR. Two new criteria are given for finding Steiner hulls—one for the Steiner tree problem on plane graphs and one for the rectilinear Steiner tree problem—which strengthen known polynomial-time methods of finding Steiner hulls.     

Two new criteria for finding Steiner hulls in Steiner tree problems

The Steiner tree problem considered in this paper is that of finding a network of minimum length connecting a given setK of terminals in a regionR of the Euclidean plane. ASteiner hull forK inR is any subregion ofR known to contain a Steiner tree forK inR. Two new criteria are given for finding Steiner hulls—one for the Steiner tree problem on plane graphs and one for the rectilinear Steiner tree problem—which strengthen known polynomial-time methods of finding Steiner hulls.This research was supported by the Air Force Office of Scientific Research under Grant AFOSR-84-0140. Reproduction in whole or part is permitted for any purpose of the United States Government.     

A note on the terminal Steiner tree problem

In 2002, Lin and Xue [Inform. Process. Lett. 84 (2002) 103-107] introduced a variant of the graph Steiner tree problem, in which each terminal vertex is required to be a leaf in the solution Steiner tree. They presented a ρ+2 approximation algorithm, where ρ is the approximation ratio of the best known efficient algorithm for the regular graph Steiner tree problem. In this note, we derive a simplified algorithm with an improved approximation ratio of 2ρ (currently ρ≈1.550, see [SODA 2000, 2000, pp. 770-790]).     

The General Steiner Tree-Star problem

The Steiner tree problem is defined as follows—given a graph G=(V,E) and a subset X⊂V of terminals, compute a minimum cost tree that includes all nodes in X. Furthermore, it is reasonable to assume that the edge costs form a metric. This problem is NP-hard and has been the study of many heuristics and algorithms. We study a generalization of this problem, where there is a “switch” cost in addition to the cost of the edges. Switches are placed at internal nodes of the tree (essentially, we may assume that all non-leaf nodes of the Steiner tree have a switch). The cost for placing a switch may vary from node to node. A restricted version of this problem, where the terminal set X cannot be connected to each other directly but only via the Steiner nodes V?X, is referred to as the Steiner Tree-Star problem. The General Steiner Tree-Star problem does not require the terminal set and Steiner node set to be disjoint. This generalized problem can be reduced to the node weighted Steiner tree problem, for which algorithms with performance guarantees of Θ(lnn) are known. However, such approach does not make use of the fact that the edge costs form a metric. In this paper we derive approximation algorithms with small constant factors for this problem. We show two different polynomial time algorithms with approximation factors of 5.16 and 5.     

Steiner Tree问题的研究进展

Steiner树问题是经典的NP难解问题,在计算机网络布局、电路设计以及生物网络等领域都有很多应用。随着参数计算理论的发展,已经证明了无向图和有向图中的Steiner树问题都是固定参数可解的(FPT)。介绍了无向图和有向图中Steiner树问题的近似算法和参数算法,分析了一些特殊Steiner树问题的研究现状,还讨论了顶点加权Steiner树问题的研究进展。最后,提出了该问题的进一步研究方向。     

欧氏Steiner最小树问题的智能优化算法

欧氏平面内连接固定原点的最小树长问题，即欧氏Steiner最小树问题，为组合优化中的NP难题，因此合理的方法是寻找启发式算法。该文给出了两种智能优化算法——模拟退火法和蚂蚁算法。首先概述智能优化算法并将中面划分成网格，然后分别介绍两种算法的原理及实现过程，最后通过一系列计算实验，测试了算法的运行性能，获得了较好的效果。     

欧氏Steiner最优树的快速算法

针对欧氏平面内连接固定原点的最小树长问题,即欧氏Steiner最优树问题,给出了插入算法、递增优化算法、遗传算法等三种快速算法,并在微机上予以实现。经大量实例测试和结果比较,获得了满意的效果。     

Concatenation-Based Greedy Heuristics for the Euclidean Steiner Tree Problem

We present a class of O(n log n) heuristics for the Steiner tree problem in the Euclidean plane. These heuristics identify a small number of subsets with few, geometrically close, terminals using minimum spanning trees and other well-known structures from computational geometry: Delaunay triangulations, Gabriel graphs, relative neighborhood graphs, and higher-order Voronoi diagrams. Full Steiner trees of all these subsets are sorted according to some appropriately chosen measure of quality. A tree spanning all terminals is constructed using greedy concatenation. New heuristics are compared with each other and with heuristics from the literature by performing extensive computational experiments on both randomly generated and library problem instances. Received October 27, 1997; revised May 7, 1998.     

Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of element-disjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k element-connected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns $\lfloor\frac{k}{2} \rfloor-1$ element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching?2.     

Approximation Algorithm for Bottleneck Steiner Tree Problem in the Euclidean Plane

A special case of thebottleneck Steiner tree problem in the Euclidean plane was considered in this paper. The problem has applications in the design of wireless communication networks, multifacility location, VLSI routing and network routing. For the special case which requires that there should be no edge connecting any two Steiner points in the optimal solution, a 3-restricted Steiner tree can be found indicating the existence of the performance ratio √2. In this paper, the special case of the problem is proved to beNP-hard and cannot be approximated within ratio √2. First a simple polynomial time approximation algorithm with performance ratio √2 is presented. Then based on this algorithm and the existence of the 3-restricted Steiner tree, a polynomial time approximation algorithm with performance ratio—√2+∈ is proposed, for any ∈>0. Supported partially by Shandong Province Excellent Middle-Aged and Young Scientists Encouragement Fund (Grant No.03BS004) and the Ministry of Education Study Abroad Returnees Research Start-up Fund, and the National Natural Science Foundation of China (Grant No.60273032).     

物流网络中节点带权的Steiner最小树的参数算法

通过优化物流的运输网络,可以有效地降低物流成本。集中配送的物流网络优化问题可以转换成求解节点带权的Steiner最小树问题,这是一个NP-hard问题。运用参数理论,提出一种新的启发式解决算法P-NSMT。算法的思想是：首先尽可能只利用终端节点构造一棵连通的最小生成树,然后逐步向树中添加能减少生成树总权值的Steiner节点,最终生成一棵节点总数不超过参数k的Steiner最小树。实验表明,与同类型其他算法相比,P-NSMT算法具有更好的准确性和时间效率,特别适应于网络规模大、终端配送节点数目较少的物流网络。     

On the terminal Steiner tree problem

We investigate a practical variant of the well-known graph Steiner tree problem. In this variant, every target vertex is required to be a leaf vertex in the solution Steiner tree. We present hardness results for this variant as well as a polynomial time approximation algorithm with performance ratio ρ+2, where ρ is the best-known approximation ratio for the graph Steiner tree problem.     

Bounding component sizes of two-connected Steiner networks

We consider the problem of constructing a shortest Euclidean 2-connected Steiner network in the plane (SMN) for a set of n terminals. This problem has natural applications in the design of survivable communication networks.In [P. Winter, M. Zachariasen, Two-connected Steiner networks: Structural properties, OR Letters 33 (2005) 395-402] we proved that all cycles in SMNs with Steiner points must have pairs of consecutive terminals of degree 2. We use this result and the notion of reduced block-bridge trees suggested by Luebke [E.L. Luebke, k-connected Steiner network problems, PhD thesis, University of North Carolina, USA, 2002] to show that no full Steiner tree in a SMN spans more than ⌊n/3⌋+1 terminals.     

瓶颈Steiner网络设计问题的算法研究

瓶颈Steiner网络设计问题要求从网络中找出一个满足某种瓶颈条件的Steiner树,由于该问题的NP困难性,因此必须找出它的近似算法。该文针对树和一般图这2种网络情形,在问题转化的基础上分别给出了基于分组Steiner问题的近似算法,在Marathe等算法思想的基础上给出了有根和无根2种情形下的2个近似算法。     

On approximation algorithms for the terminal Steiner tree problem

The terminal Steiner tree problem is a special version of the Steiner tree problem, where a Steiner minimum tree has to be found in which all terminals are leaves. We prove that no polynomial time approximation algorithm for the terminal Steiner tree problem can achieve an approximation ratio less than (1−o(1))lnn unless NP has slightly superpolynomial time algorithms. Moreover, we present a polynomial time approximation algorithm for the metric version of this problem with a performance ratio of 2ρ, where ρ denotes the best known approximation ratio for the Steiner tree problem. This improves the previously best known approximation ratio for the metric terminal Steiner tree problem of ρ+2.     

基于设施选址的Steiner问题的算法

在设施选址问题的基础上给出了广义Steiner树-星问题的两个近似比分别为3．55和3．582的近似算法,并在问题转化的基础上研究了其他若干特殊情形的Steiner树问题的近似算法。     

Local Optimization in the Steiner Problem on the Euclidean Plane

By the local optimal Steiner tree is meant a tree with optimally distributed Steiner points for a given adjacency matrix. The adjacency matrix defines the point of local minimum, and all arrangements (coordinates) of the Steiner points that are admissible for it define the minimum neighborhood. Solution is local optimal if the tree length cannot be reduced by rearranging the Steiner points. An algorithm of local optimization based on the concept of coordinatewise descent was considered.     

The Steiner Problem in Graphs: Topological Methods of Solution

Algorithms for the Steiner problem and its generalizations on large graphs with a relatively small number of terminal vertices are designed by a two-level solution scheme: a network topology (i.e., a tree determining the adjacency of terminal vertices and branching points) is determined in the upper level and the optimal location for branching points with the topology found in the upper level is determined in the lower level.     

图的steiner最小树问题及其求解

斯坦纳树问题是组合优化学科中的一个问题。属于NP-难问题,即无法在多项式时间内得到最优解。本文主要讨论了图的steiner最小树问题,并给出了近似算法,该算法是在破圈法的基础上进行了改进,并且引用了agent的思想。最后对算法进行了分析。     

The class Steiner minimal tree problem: a lower bound and test problem generation

The class Steiner minimal tree problem is an extension of the standard Steiner minimal tree problem in graphs, motivated by the problem of wire routing in the area of physical design of very large scale integration (VLSI). This problem is NP-hard, even if there are no Steiner nodes and the graph is a tree; moreover, there exists no polynomial time approximation algorithm with a constant bound on the relative error under the hypothesis that P NP [16]. Hence, fast and good heuristic algorithms are needed in practice. In this paper, we present an integer programming formulation of the problem. Using Lagrangean relaxation and subgradient optimization, we derive a lower bound. In order to test the lower bound, we present a procedure for generating test problems for the class Steiner minimal tree problem that have known optimal solutions. The computational experiments for the test problems demonstrate that the lower bound is very tight and differs from the optimal solutions by only a few percent on average for sparse graphs. Received: 5 July 1999 / Revised version: 14 July 2000