The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

Optimization in Telecommunication Networks

Network design and network synthesis have been the classical optimization problems intelecommunication for a long time. In the recent past, there have been many technologicaldevelopments such as digitization of information, optical networks, internet, and wirelessnetworks. These developments have led to a series of new optimization problems. Thismanuscript gives an overview of the developments in solving both classical and moderntelecom optimization problems.We start with a short historical overview of the technological developments. Then,the classical (still actual) network design and synthesis problems are described with anemphasis on the latest developments on modelling and solving them. Classical results suchas Menger’s disjoint paths theorem, and Ford-Fulkerson’s max-flow-min-cut theorem, butalso Gomory-Hu trees and the Okamura-Seymour cut-condition, will be related to themodels described. Finally, we describe recent optimization problems such as routing andwavelength assignment, and grooming in optical networks.operations research and management science;

A Potential-Field-Based Multilevel Algorithm for Drawing Large Graphs

The aim of automatic graph drawing is to compute a well-readable layout of a given graph G=(V,E). One very popular class of algorithms for drawing general graphs are force-directed methods. These methods generate drawings of G in the plane so that each edge is represented by a straight line connecting its two adjacent nodes. The computation of the drawings is based on associating G with a physical model. Then, the algorithms iteratively try to find a placement of the nodes so that the total energy of the physical system is minimal. Several force-directed methods can visualize large graphs containing many thousands of vertices in reasonable time. However, only some of these methods guarantee a sub-quadratic running time in special cases or under certain assumptions, but not in general. The others are not sub-quadratic at all. We develop a new force-directed algorithm that is based on a combination of an efficient multilevel strategy and a method for approximating the repulsive forces in the system by rapidly evaluating potential fields. The worst-case running time of the new method is O(|V| log|V|+|E|) with linear memory requirements. In practice, the algorithm generates nice drawings of graphs containing up to 100000 nodes in less than five minutes. Furthermore, it clearly visualizes even the structures of those graphs that turned out to be challenging for other tested methods

Antenna-coupled TES bolometer arrays for CMB polarimetry

We describe the design and performance of polarization selective antenna-coupled TES arrays that will be used in several upcoming Cosmic Microwave Background (CMB) experiments: SPIDER, BICEP-2/SPUD. The fully lithographic polarimeter arrays utilize planar phased-antennas for collimation (F/4 beam) and microstrip filters for band definition (25% bandwidth). These devices demonstrate high optical efficiency, excellent beam shapes, and well-defined spectral bands. The dual-polarization antennas provide well-matched beams and low cross polarization response, both important for high-fidelity polarization measurements. These devices have so far been developed for the 100 GHz and 150 GHz bands, two premier millimeter-wave atmospheric windows for CMB observations. In the near future, the flexible microstrip-coupled architecture can provide photon noise-limited detection for the entire frequency range of the CMBPOL mission. This paper is a summary of the progress we have made since the 2006 SPIE meeting in Orlando, FL

Antenna-coupled TES bolometer arrays for CMB polarimetry

We describe the design and performance of polarization selective antenna-coupled TES arrays that will be used in several upcoming Cosmic Microwave Background (CMB) experiments: SPIDER, BICEP-2/SPUD. The fully lithographic polarimeter arrays utilize planar phased-antennas for collimation (F/4 beam) and microstrip filters for band definition (25% bandwidth). These devices demonstrate high optical efficiency, excellent beam shapes, and well-defined spectral bands. The dual-polarization antennas provide well-matched beams and low cross polarization response, both important for high-fidelity polarization measurements. These devices have so far been developed for the 100 GHz and 150 GHz bands, two premier millimeter-wave atmospheric windows for CMB observations. In the near future, the flexible microstrip-coupled architecture can provide photon noise-limited detection for the entire frequency range of the CMBPOL mission. This paper is a summary of the progress we have made since the 2006 SPIE meeting in Orlando, FL