We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, which are unique connected solutions of the Steiner problem for some given finite sets of...

Abstract: Steiner triple systems are among the simplest and most intensively studied combinatorial designs. Their origins go back to the 1840s, and there exists by now a sizeable literature on the topic. In 1980, Babai proved that almost all Steiner triple systems have no nontrivial automorphism. On the other hand, there exist Steiner triple systems with large automorphism groups. We will discu...

The Euclidean Steiner tree problem is solved by finding the tree with minimal Euclidean length spanning a set of fixed vertices in the plane, while allowing for the addition of auxiliary vertices (Steiner vertices). Steiner trees are widely used to design real-world structures like highways and oil pipelines. Unfortunately, the Euclidean Steiner Tree Problem has shown to be NP-Hard, meaning the...

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimal possible total length. The Steiner ratio for a metric space is the largest lower bound for the ratio of lengths between a minimum Steiner tree and a minimum spanning tree on the same set of points in the metric space. Du et al. (1993) conjectured that the Steiner ratio on a normed plan...

Following the entropy method this paper presents general concentration inequalities, which can be applied to combinatorial optimization and empirical processes. The inequalities give improved concentration results for optimal travelling salesmen tours, Steiner trees and the eigenvalues of random symmetric matrices. 1 Introduction Since its appearance in 1995 Talagrand’s convex distance inequali...

Recently D. Buchholz and R. Verch have proposed a method for implementing in algebraic quantum field theory ideas from renormalization group analysis of short-distance (high energy) behavior by passing to certain scaling limit theories. Buchholz and Verch distinguish between different types of theories where the limit is unique, degenerate, or classical, and the method allows in principle to ex...

We study the approximability of three versions of the Steiner tree problem. For the rst one where the input graph is only supposed connected, we show that it is not approximable within better than |V \N |−2 for any 2 ∈ (0, 1), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where ...

In the rectilinear Steiner arborescence problem the task is to build a shortest rectilinear Steiner tree connecting a given root and a set of terminals which are placed in the plane such that all root-terminal-paths are shortest paths. This problem is known to be NP-hard. In this paper we consider a more restricted version of this problem. In our case we have a depth restrictions d(t) ∈ N for e...

We discuss a multiple genome rearrangement problem by signed reversals: Given a collection of genomes, we generate them in the minimum number of signed reversals. It is NP-hard and equivalent to finding an optimal Steiner tree to connect the genomes by reversal paths. We design two algorithms to find the optimal Steiner nodes of the problem: Neighbor-perturbing algorithm and branch-and-bound al...

Let P be a set of n points in a metric space. A Steiner Minimal Tree (SMT) on P is a shortest network interconnecting P while a Minimum Spanning Tree (MST) is a shortest network interconnecting P with all edges between points of P . The Steiner ratio is the infimum over P of ratio of the length of SMT over that of MST. Steiner ratio problem is to determine the value of the ratio. In this paper ...