On better heuristics for Steiner minimum trees

Finding a shortest network interconnecting a given set of points in a metric space is called the Steiner minimum tree problem. The Steiner ratio is the largest lower bound for the ratio between lengths of a Steiner minimum tree and a minimum spanning tree for the same set of points. In this paper, we show that in a metric space, if the Steiner ratio is less than one and finding a Steiner minimum tree for a set of size bounded by a fixed number can be performed in polynomial time, then there exists a polynomialtime heuristic for the Steiner minimum tree problem with performance ratio bigger than the Steiner ratio. It follows that in the Euclidean plane, there exists a polynomial-time heuristic for Steiner minimum trees with performance ratio bigger than . This solves a long-standing open problem.Part of this work was done while this author visited the Department of Computer Science, Princeton University, supported in part by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648, supported in part by NSF grant No. CCR-8920505, and also supported in part by the National Natural Science Foundation of China.     

Wiener Index of Trees: Theory and Applications

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.     

A Class of Trees and its Wiener Index

In this paper, we will consider the Wiener index for a class of trees that is connected to partitions of integers. Our main theorem is the fact that every integer is the Wiener index of a member of this class. As a consequence, this proves a conjecture of Lepović and Gutman. The paper also contains extremal and average results on the Wiener index of the studied class.This work was supported by Austrian Science Fund project no. S-8307-MAT.     

Equiseparability on terminal Wiener index

The aim of this work is to explore the properties of the terminal Wiener index, which was recently proposed by Gutman et al. (2004) [3], and to show the fact that there exist pairs of trees and chemical trees which cannot be distinguished by using it. We give some general methods for constructing equiseparable pairs and compare the methods with the case for the Wiener index. More specifically, we show that the terminal Wiener index is degenerate to some extent.     

Laplacian coefficients of trees with given number of leaves or vertices of degree two

Let G be a simple undirected n-vertex graph with the characteristic polynomial of its Laplacian matrix . It is well known that for trees the Laplacian coefficient c_n-2 is equal to the Wiener index of G, while c_n-3 is equal to the modified hyper-Wiener index of graph. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize the trees with k leaves (pendent vertices) which simultaneously minimize all Laplacian coefficients. In particular, this extremal balanced starlike tree S(n,k) minimizes the Wiener index, the modified hyper-Wiener index and recently introduced Laplacian-like energy. We prove that graph S(n,n-1-p) has minimal Laplacian coefficients among n-vertex trees with p vertices of degree two. In conclusion, we illustrate on examples of these spectrum-based invariants that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution, and pose a conjecture on extremal unicyclic graphs with k leaves.     

带梯子的曲折线上的最短树问题

Abstract. In this paper,Steiner minimal trees for point sets with special structure are studied.These sets consist of zigzag lines and equidistant points lying on them.     

Approximations for Steiner Trees with Minimum Number of Steiner Points

Given n terminals in the Euclidean plane and a positive constant, find a Steiner tree interconnecting all terminals with the minimum number of Steiner points such that the Euclidean length of each edge is no more than the given positive constant. This problem is NP-hard with applications in VLSI design, WDM optical networks and wireless communications. In this paper, we show that (a) the Steiner ratio is 1/ 4, that is, the minimum spanning tree yields a polynomial-time approximation with performance ratio exactly 4, (b) there exists a polynomial-time approximation with performance ratio 3, and (c) there exists a polynomial-time approxi-mation scheme under certain conditions.     

Analysis of Steiner subtrees of random trees for traceroute algorithms

We consider in this article the problem of discovering, via a traceroute algorithm, the topology of a network, whose graph is spanned by an infinite branching process. A subset of nodes is selected according to some criterion. As a measure of efficiency of the algorithm, the Steiner distance of the selected nodes, i.e. the size of the spanning subtree of these nodes, is investigated. For the selection of nodes, two criteria are considered: a node is randomly selected with a probability, which is either independent of the depth of the node (uniform model) or else in the depth biased model, is exponentially decaying with respect to its depth. The limiting behavior the size of the discovered subtree is investigated for both models. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

固定直径的树的Wiener指数

图G的wiener指数定义为图中所有点对u,v的距离之和∑d（u,v）. 在这篇文章中,我们刻画了在n个顶点直径为d的所有树中具有第三小wiener指数的树的特征以及介绍了得到这类树的wiener指数排序的方法.     

A probably fast,provably optimal algorithm for rectilinear Steiner trees

We use the technique of divide-and-conquer to construct a rectilinear Steiner minimal tree on a set of sites in the plane. A well-known optimal algorithm for this problem by Dreyfus and Wagner [10] is used to solve the problem in the base case. The run time of our optimal algorithm is probabilistic in nature: for all ? > 0, there exists b > 0 such that Prob[T(n) > 2^{b√n log n]>1–?, for n log n} > 1 – ?, for n sites uniformly distributed on a rectangle. The key fact in the run-time argument is the existence of probable bounds on the number of edges of an optimal tree crossing our subdivision lines. We can test these bounds in low-degree polynomial time for any given set of sites. © 1994 John Wiley & Sons, Inc.     

Integrality gap of the hypergraphic relaxation of Steiner trees: A short proof of a 1.55 upper bound

Recently, Byrka, Grandoni, Rothvoß and Sanità gave a 1.39 approximation for the Steiner tree problem, using a hypergraph-based linear programming relaxation. They also upper-bounded its integrality gap by 1.55. We describe a shorter proof of the same integrality gap bound, by applying some of their techniques to a randomized loss-contracting algorithm.     

A sharp lower bound for the first eigenvalue on Finsler manifolds

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.