The rectilinear Steiner tree problem requires a shortest tree spanning a given vertex subset in the plane with rectilinear distance. It was proved that the output length of Zelikovsky's 25] and Berman/Ramaiyer 3] heuristics is at most 1.375 and 97 72 1:347 of the optimal length, respectively. It was claimed that these bounds are not tight. Here we improve these bounds to 1.3125 and 61 48 1:271,...

For any proper block colouring of a Steiner system, a palette of an element is the set of colours on blocks incident with it. We obtain bounds on the minimum possible number of distinct palettes in proper block colourings of Steiner triple systems and Steiner systems S(2, 4, v).

Was ist ein minimaler Spannbaum? Welche Algorithmen zur Berechnung minimaler Spannbäume kennen Sie? Wie funktionieren diese Algorithmen und welche Laufzeit haben sie? Was ist das Steiner-Problem? Für welche Anwendungen ist das Steiner-Problem relevant? Wie funktioniert die Distanz-Heuristik zur Berechnung von Steiner-Bäumen? Welche Approximationsrate erzielt die Distanz-Heuristik und wie kann m...

The Steiner tree problem requires to nd a shortest tree connecting a given set of terminal points in a metric space. We suggest a better and fast heuristic for the Steiner problem in graphs and in rectilinear plane. This heuristic nds a Steiner tree at most 1.757 and 1.267 times longer than the optimal solution in graphs and rectilinear plane, respectively.

We consider a reliable network design problem under uncertain edge failures. Our goal is to select a minimum-cost subset of edges in the network to connect multiple terminals together with high probability. This problem can be seen as a stochastic variant of the Steiner tree problem. We propose two scenario-based Steiner cut formulations, study the strength of the proposed valid inequalities, a...

Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance differe...

Two Steiner triple systems, each containing precisely one Pasch configuration which, when traded, switches one system to the other, are called twin Steiner triple systems. If the two systems are isomorphic the systems are called identical twins. Hitherto, identical twins were only known for orders 21, 27 and 33. In this paper we construct infinite families of identical twin Steiner triple systems.

We consider the Shallow-Light Steiner Network problem from a fixed-parameter perspective. Given a graph G, a distance bound L, and p pairs of vertices (s1, t1), . . . , (sp, tp), the objective is to find a minimum-cost subgraph G′ such that si and ti have distance at most L in G′ (for every i ∈ [p]). Our main result is on the fixed-parameter tractability of this problem with parameter p. We exa...

Motivated by applications in combinatorial group testing for consecutive positives, we investigate a block sequence of a maximum packing MP(t, k, v) which contains the blocks exactly once such that the collection of all blocks together with all unions of two consecutive blocks of this sequence forms an error correcting code with minimum distance d. Such a sequence is usually called a block sequ...

Among other results, we prove the following theorem about Steiner minimal trees in ddimensional Euclidean space: if two finite sets in R have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.