The Steiner tree problem is a classical, well-studied, NP-hard optimization problem. Here we are given an undirected graph G = (V,E), a subset R of V of terminals, and nonnegative costs ce for all edges e in E. A feasible Steiner tree for a given instance is a tree T in G that spans all terminals in R. The goal is to compute a feasible Steiner tree of smallest cost. In this thesis we will focus...

The notion of Steiner visibility graphs is introduced. Their applicability in connection with the construction of good quality suboptimal solutions to the Euclidean Steiner tree problem with obstacles is discussed. Polynomial algorithms generating Steiner visibility graphs are described.

Analyzing the sub-level sets of the distance to a compact sub-manifold of R d is a common method in TDA to understand its topology. The distance to measure (DTM) was introduced by Chazal, Cohen-Steiner and M{\'e}rigot in [7] to face the non-robustness of the distance to a compact set to noise and outliers. This function makes possible the inference of the topology of a compact subset of R d fro...

Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g + 1 with minimum Hamming distance 2k − 3, in which each codeword has length v and weight k. As to the existence of a GS(2, k, v, g), a lot of work has been done for k = 3, while not so much is known for k = 4. The notion k-GDD was first introd...

In this paper we provide an approximation à la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the ...

In this paper we prove the existence of certain Steiner systems. The results of this paper are divided into three sections: In section 1, we investigate Steiner systems of type S(3, 4, 8). In section 2, we study Steiner systems of type S(5, 6, 12), and in the last section we study Steiner systems of type (5, 8, 24), the MOG and the Hexacode. Mathematics Subject Classification: 20C15, 20E28

It is shown that a class of Steiner triple systems of order 2−1, obtained by some special switchings from the Hamming Steiner triple system, is embedded into some perfect code, constructed by known switchings of ijk-components from the binary Hamming code. The number of Steiner triple systems of order n and rank less or equal n− log(n + 1) + 2, embedded into perfect binary codes of length n, is...