Steiner triple systems are known to exist for orders n ≡ 1, 3 mod 6, the admissible orders. There are many known constructions for infinite classes of Steiner triple systems. However, Steiner triple systems that lack prescribed configurations are harder to find. This paper gives a proof that the spectrum of orders of 5-sparse Steiner triple systems has arithmetic density 1 as compared to the ad...

Given a directed graph G and a list (s1, t1), . . . , (sk, tk) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed si → ti path for every 1 ≤ i ≤ k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, . . . , tk) is known to be fixed-parameter tractable parameterized by the number of term...

E. J. Hughes, D. J. Sharp, D. L. Hill, A. Wink, J. V. Hajnal, and S. J. Counsell Robert Steiner MRI Unit, Hammersmith Hospital, Imperial College, London, England, United Kingdom, Cognitive Neuroimaging Group, MRC Clinical sciences Centre, Imperial college, London, England, United Kingdom, The London Bioscience Innovation Centre, IXICO ltd, London, England, Hammersmith Hospital, Imperial College...

We initiate a systematic study of embeddings of Steiner triple systems into Steiner systems S(2; 4; v). We settle the existence of an embedding of the unique STS(7) and, with one possible exception, of the unique STS(9) into S(2; 4; v). We also obtain bounds for embedding sizes of Steiner triple systems of larger orders. c © 2003 Elsevier B.V. All rights reserved.

This paper studies a 4-approximation algorithm for k-prize collecting Steiner tree problems. This problem generalizes both k-minimum spanning tree problems and prize collecting Steiner tree problems. Our proposed algorithm employs two 2-approximation algorithms for k-minimum spanning tree problems and prize collecting Steiner tree problems. Also our algorithm framework can be applied to a speci...

We will be studying the Group Steiner tree problem in this lecture. Recall that the classical Steiner tree problem is the following. Given a weighted graphG = (V,E), a subset S ⊆ V of the vertices, and a root r ∈ V , we want to find a minimum weight tree which connects all the vertices in S to r. The weights on the edges are assumed to be positive. We will now define the Group Steiner tree prob...

In this thesis we will attempt to close the gap between fast grid-based method and -approximation methods that solve the Weighted Region problem. In order to compare both methods, we will define a formula to calculate the -value of grid-based methods and define the bounds of the -value for -approximation methods. Furthermore, we will improve on heuristics for A* grid methods for weighted region...