Approximation Algorithms for Generalized MST and TSP in Grid Clusters

We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each grid cell is $1 \times 1$. In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a $(1+4\sqrt{2}+\epsilon)$ and $(1.5+8\sqrt{2}+\epsilon)$-approximation algorithm for these two problems in the described setting, respectively. Our motivation is based on the problem posed in [7] for a constant approximation algorithm. The authors designed a PTAS for the more special case of the GMST where non-empty cells are connected end dense enough. However, their algorithm heavily relies on this connectivity restriction and is unpractical. Our results develop the topic further

Evaluation des besoins et des activités. Problématique de leur localisation. Critères, méthodes et applications pour la révision des plans de secteur. Problématique pour la mise en oeuvre des zones d'aménagement différé

Rapport final de la subvention 2001, Cinquième Volum

Collaborative European NeuroTrauma Effectiveness Research in Traumatic Brain Injury (CENTER-TBI): A Prospective Longitudinal Observational Study

BACKGROUND: Current classification of traumatic brain injury (TBI) is suboptimal, and management is based on weak evidence, with little attempt to personalize treatment. A need exists for new precision medicine and stratified management approaches that incorporate emerging technologies. OBJECTIVE: To improve characterization and classification of TBI and to identify best clinical care, using comparative effectiveness research approaches. METHODS: This multicenter, longitudinal, prospective, observational study in 22 countries across Europe and Israel will collect detailed data from 5400 consenting patients, presenting within 24 hours of injury, with a clinical diagnosis of TBI and an indication for computed tomography. Broader registry-level data collection in approximately 20 000 patients will assess generalizability. Cross sectional comprehensive outcome assessments, including quality of life and neuropsychological testing, will be performed at 6 months. Longitudinal assessments will continue up to 24 months post TBI in patient subsets. Advanced neuroimaging and genomic and biomarker data will be used to improve characterization, and analyses will include neuroinformatics approaches to address variations in process and clinical care. Results will be integrated with living systematic reviews in a process of knowledge transfer. The study initiation was from October to December 2014, and the recruitment period was for 18 to 24 months. EXPECTED OUTCOMES: Collaborative European NeuroTrauma Effectiveness Research in TBI should provide novel multidimensional approaches to TBI characterization and classification, evidence to support treatment recommendations, and benchmarks for quality of care. Data and sample repositories will ensure opportunities for legacy research. DISCUSSION: Comparative effectiveness research provides an alternative to reductionistic clinical trials in restricted patient populations by exploiting differences in biology, care, and outcome to support optimal personalized patient management

The geometric generalized minimum spanning tree problem with grid clustering

This paper is concerned with a special case of the generalized minimum spanning tree problem. The problem is defined on an undirected graph, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges. The problem is to find a tree of minimum cost containing at least one vertex in each cluster. We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance defines the edge cost. We prove that the problem is strongly -hard even in the case of a special structure of the clustering called grid clustering. We construct an exact exponential time dynamic programming algorithm and, based on this dynamic programming algorithm, we develop a polynomial time approximation scheme for the problem with grid clustering