Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems

We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum. An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weighted matching yields a polynomial solution for the MWMP, the practical behavior is just the opposite, and we can solve the FWP with high accuracy in order to find a good heuristic solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental Algorithms, 200

GCIP water and energy budget synthesis (WEBS)

As part of the World Climate Research Program\u27s (WCRPs) Global Energy and Water-Cycle Experiment (GEWEX) Continental-scale International Project (GCIP), a preliminary water and energy budget synthesis (WEBS) was developed for the period 1996–1999 from the “best available” observations and models. Besides this summary paper, a companion CD-ROM with more extensive discussion, figures, tables, and raw data is available to the interested researcher from the GEWEX project office, the GAPP project office, or the first author. An updated online version of the CD-ROM is also available at http://ecpc.ucsd.edu/gcip/webs.htm/. Observations cannot adequately characterize or “close” budgets since too many fundamental processes are missing. Models that properly represent the many complicated atmospheric and near-surface interactions are also required. This preliminary synthesis therefore included a representative global general circulation model, regional climate model, and a macroscale hydrologic model as well as a global reanalysis and a regional analysis. By the qualitative agreement among the models and available observations, it did appear that we now qualitatively understand water and energy budgets of the Mississippi River Basin. However, there is still much quantitative uncertainty. In that regard, there did appear to be a clear advantage to using a regional analysis over a global analysis or a regional simulation over a global simulation to describe the Mississippi River Basin water and energy budgets. There also appeared to be some advantage to using a macroscale hydrologic model for at least the surface water budgets

Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes

Magnetic properties of pure and Gd doped EuO probed by NMR

An Eu NMR study in the ferromagnetic phase of pure and Gd doped EuO was performed. A complete description of the NMR lineshape of pure EuO allowed for the influence of doping EuO with Gd impurities to be highlighted. The presence of a temperature dependent static magnetic inhomogeneity in Gd doped EuO was demonstrated by studying the temperature dependence of the lineshapes. The results suggest that the inhomogeneity in 0.6% Gd doped EuO is linked to colossal magnetoresistance. The measurement of the spin-lattice relaxation times as a function of temperature led to the determination of the value of the exchange integral J as a function of Gd doping. It was found that J is temperature independent and spatially homogeneous for all the samples and that its value increases abruptly with increasing Gd doping.Comment: 14 pages, 10 figures, to be published in Physical Review

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.Comment: 26 pages. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop "Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems" held in February 2015. Comments are welcome. Revised version corrects some typos, updates the references, and clarifies some hypotheses. To appear in the Journal of Algebraic Combinatoric

Problem solving ability and repetition of deliberate self-harm: a multicentre study.

Background. While recent studies have found problem-solving impairments in individuals who engage in deliberate self-harm (DSH), few studies have examined repeaters and non-repeaters separately. The aim of the present study was to investigate whether specific types of problem-solving are associated with repeated DSH. Method. As part of the WHO/EURO Multicentre Study on Suicidal Behaviour, 836 medically treated DSH patients (59% repeaters) from 12 European regions were interviewed using the European Parasuicide Study Interview Schedule (EPSIS II) approximately 1 year after their index episode. The Utrecht Coping List (UCL) assessed habitual responses to problems. Results. Factor analysis identified five dimensions – Active Handling, Passive-Avoidance, Problem Sharing, Palliative Reactions and Negative Expression. Passive-Avoidance – characterized by a pre-occupation with problems, feeling unable to do anything, worrying about the past and taking a gloomy view of the situation, a greater likelihood of giving in so as to avoid difficult situations, the tendency to resign oneself to the situation, and to try to avoid problems – was the problem-solving dimension most strongly associated with repetition, although this association was attenuated by self-esteem. Conclusions. The outcomes of the study indicate that treatments for DSH patients with repeated episodes should include problem-solving interventions. The observed passivity and avoidance of problems (coupled with low self-esteem) associated with repetition suggests that intensive therapeutic input and follow-up are required for those with repeated DSH

A unique haplotype of RCCX copy number variation: from the clinics of congenital adrenal hyperplasia to evolutionary genetics.

There is a difficulty in the molecular diagnosis of congenital adrenal hyperplasia (CAH) due to the c.955C>T (p.(Q319*), formerly Q318X, rs7755898) variant of the CYP21A2 gene. Therefore, a systematic assessment of the genetic and evolutionary relationships between c.955C>T, CYP21A2 haplotypes and the RCCX copy number variation (CNV) structures, which harbor CYP21A2, was performed. In total, 389 unrelated Hungarian individuals with European ancestry (164 healthy subjects, 125 patients with non-functioning adrenal incidentaloma and 100 patients with classical CAH) as well as 34 adrenocortical tumor specimens were studied using a set of experimental and bioinformatic methods. A unique, moderately frequent (2%) haplotypic RCCX CNV structure with three repeated segments, abbreviated to LBSASB, harboring a CYP21A2 with a c.955C>T variant in the 3'-segment, and a second CYP21A2 with a specific c.*12C>T (rs150697472) variant in the middle segment occurred in all c.955C>T carriers with normal steroid levels. The second CYP21A2 was free of CAH-causing mutations and produced mRNA in the adrenal gland, confirming its functionality and ability to rescue the carriers from CAH. Neither LBSASB nor c.*12C>T occurred in classical CAH patients. However, CAH-causing CYP21A2 haplotypes with c.955C>T could be derived from the 3'-segment of LBSASB after the loss of functional CYP21A2 from the middle segment. The c.*12C>T indicated a functional CYP21A2 and could distinguish between non-pathogenic and pathogenic genomic contexts of the c.955C>T variant in the studied European population. Therefore, c.*12C>T may be suitable as a marker to avoid this genetic confound and improve the diagnosis of CAH