Classifying Tractable Instances of the Generalized Cable-Trench Problem

Given a graph $G$ rooted at a vertex $r$ and weight functions, $\gamma , \tau: E(G) \to \mathbb{R}$, the generalized cable-trench problem (CTP) is to find a single spanning tree that simultaneously minimizes the sum of the total edge cost with respect to $\tau$ and the single-source shortest paths cost with respect to $\gamma$. Although this problem is provably $NP$-complete in the general case, we examine certain tractable instances involving various graph constructions of trees and cycles, along with quantities associated to edges and vertices that arise out of these constructions. We show that given a path decomposition oracle, for graphs in which all cycles are edge disjoint, there exists a fast method to determine the cable-trench tree. Further, we examine properties of graphs which contribute to the general intractability of the CTP and present some open questions in this direction

Low 25-OH vitamin d is associated with benign prostatic hyperplasia

Purpose: We tested the hypothesis that low vitamin D is associated with benign prostatic hyperplasia. We also studied whether body composition, sex hormones, serum sex hormone-binding globulin, albumin corrected serum calcium, adiponectin and lipid status are associated with benign prostatic hyperplasia. Materials and Methods: We investigated 184 representative, randomly selected men 72 to 76 years old enrolled in the Gothenburg arm of the Osteoporotic Fractures in Men Study (MrOS). Men with a history of prostate cancer, prostate operation or medication for benign prostatic hyperplasia were excluded from study, leaving 155 available for analysis. A cross-sectional study was performed in which benign prostatic hyperplasia measured by total prostate volume was related to clinical, anthropometric, endocrine and metabolic factors on univariate and multivariate analyses with regression models. Results: Median prostate volume was 40 ml. In multivariate models only 25-OH vitamin D, albumin corrected serum calcium, serum sex hormone-binding globulin and high density lipoprotein cholesterol were significantly and inversely associated with large prostate glands. Conclusions: The current report adds 4 independent factors associated with benign prostatic hyperplasia, including low 25-OH vitamin D, serum calcium, sex hormone-binding globulin and high density lipoprotein cholesterol