In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1, 2, . . . ,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97 ·n · logW ) using O(n · lognW · log k) space. This is the first single-exponential time, polynomial-sp...

Given a set of i (i = 1; 2; : : :; k) orientations (angles) in the plane, one can deene a distance function which induces a metric in the plane, called the orientation metric 3]. In the special case where all the angles are equal, we call the metric a uniform orientation metric 2]. Speciically, if there are orientations, forming angles ii ; 0 i ?1, with the x-axis, where 2 is an integer, we cal...

We consider the currently strongest Steiner tree approximation algorithm that has recently been published by Goemans, Olver, Rothvoß and Zenklusen (2012). It first solves a hypergraphic LP relaxation and then applies matroid theory to obtain an integral solution. The cost of the resulting Steiner tree is at most (1.39 + ε)-times the cost of an optimal Steiner tree where ε tends to zero as some ...

In the Steiner Networks Activation problem we are given a graph G = (V,E), S ⊆ V , a family {fuv(xu, xv) : uv ∈ E} of monotone non-decreasing activating functions from R2+ to {0, 1} each, and connectivity requirements {r(u, v) : u, v ∈ V }. The goal is to find a weight assignment w = {wv : v ∈ V } of minimum total weight w(V ) = ∑ v∈V wv, such that: for all u, v ∈ V , the activated graph Gw = (...

This paper defines a class of designs which generalise t-designs, resolvable designs, and orthogonal arrays. For the parameters t = 2, k = 3 and λ = 1, the designs in the class consist of Steiner triple systems, Latin squares, and 1-factorisations of complete graphs. For other values of t and k, we obtain t-designs, Kirkman systems, large sets of Steiner triple systems, sets of mutually orthogo...

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G = (V, E) and an integer k, decide whether there exists F ⊆ V , |F | ≤ k, such that G[V \ F ] is a forest and G[F ] is connected. We show that Connected ...

In [4] we introduced a product construction for cyclic Steiner quadruple systems. The purpose of this paper is to modify the construction so that it is applicable to cyclic Steiner 2-designs. A cyclic Steiner 2-design is a Steiner system s(t, k, u), 2 d t < k < u, in which t = 2 and which has an automorphism of order u. We denote such a system by CS(2, k, u). Under suitable conditions we show h...

We consider the Steiner Multicut problem, which asks, given an undirected graph G, a collection T = {T1, . . . , Tt}, Ti ⊆ V (G), of terminal sets of size at most p, and an integer k, whether there is a set S of at most k edges or nodes such that of each set Ti at least one pair of terminals is in different connected components of G \S. This problem generalizes several well-studied graph cut pr...

The Steiner tree problem which is known to be NP-Complete is the following. Given a weighted undirected graph G = (V;E), and a set X V of terminals, the objective is to nd a tree of minimum cost which connects all the terminals. If the graph is directed, in addition to X, we are given a root r 2 V , and the objective is to nd a minimum cost arborescence which connects the root to each of the te...

Let G be a graph. The Steiner distance of $$W\subseteq V(G)$$ is the minimum size connected subgraph containing W. Such necessarily tree called W-tree. set $$A\subseteq k-Steiner general position if $$V(T_B)\cap A = B$$ holds for every $$B\subseteq A$$ cardinality k, and B-tree $$T_B$$ . number $$\mathrm{sgp}_k(G)$$ largest in G. cliques are introduced used to bound from below. determined trees...