When relating a set of sequences by a phylogeny, we are essentially constructing a Steiner tree connecting the sequences in the space of all finite sequences. Finding an optimal Steiner tree is in most formulations hard, so population genetics and phylogenetics have often used spanning trees as an approximation for computational expediency. In this assessment you will be asked to investigate an...

1 Overview During this lecture we will present approximation algorithms for three network design problems: Group Steiner Tree, k-median and k-center. Goal: Find a tree in G that connects at least one vertex from each S i. Observe that we may guess one vertex r ∈ V (e.g. by trying all vertices from S 1), that is a part of an optimum solution. We want to prove that if G is a tree rooted at r of d...

The Ramsey number r(H) of a graph H is the minimum integer n such that any two-coloring edges complete Kn contains monochromatic copy H. While this definition only asks for single H, it often case every two-edge-coloring on vertices many copies over all two-colorings Kr(H) will be referred to as threshold multiplicity Addressing problem Harary and Prins, who were first systematically study quan...

Lydia Steiner Bioinformatics Group, Interdisciplinary Center for Bioinformatics, University of Leipzig ∗ Peter F. Stadler Bioinformatics Group, University of Leipzig, Interdisciplinary Center for Bioinformatics, University of Leipzig; Max-Planck-Institute for Mathematics in the Science; Fraunhofer Institute for Cell Therapy and Immunology; Center for non-coding RNA in Technology and Health, Uni...

The stability of a communication network composed of processing nodes and communication links is of prime importance to network designers. As the network begins losing links or nodes, eventually there is a loss in its effectiveness. Thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible...

In communication networks, many applications, such as video on demand and video conferencing, must establish a communications tree that spans a subset K in a vertex set. The source node can then send identical data to all nodes in set K along this tree. This kind of communication is known as multicast communication. A network optimization problem, called the Steiner tree problem (STP), is prese...

one of the most critical issues in designing Wireless Sensor Network (WSN) is to minimize the energy consumption. In Wireless Sensor Networks, data aggregation reduces the redundancy among sensed data and optimal sensor routing algorithm provides strategy for data gathering with minimum energy. The energy consumption is reduced by combining data fusion and cluster based routing. In this paper, ...

in accordance with the theorem in Katz [1955]. This result was anticipated in turn by Rubin and Sitgreaves in an unpublished memorandum cited in Katz [1955]. In order to contribute something positive in this note, we now prove the theorem about graphs and groups which justifies the formula given in Harary and Read [1966] for the number of ways to label a structure. Since this is a sequel to Har...

We count the number of nonisomorphic geometric minimum spanning trees formed by adding a single point to an n-point set in d-dimensional space, by relating it to a family of convex decompositions of space. The O(n log 2−d n) bound that we obtain significantly improves previously known bounds and is tight to within a polylogarithmic factor. The minimum Steiner tree of a point set is the tree of ...

We demonstrate that ` rounds of the Sherali-Adams hierarchy and 2` rounds of the Lovász-Schrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in `-layered graphs from Ω( √ k) to O(` · log k) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2` rounds of the considerably stronger Lasserre SDP hierarchy r...