Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees

Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a $\tilde{O}(mn)$ time algorithm that computes a spanning tree of degree at most $\Delta^* +1$ is previously known [F\"urer \& Raghavachari 1994]; here $\Delta^*$ denotes the minimum tree degree of all the spanning trees. In this paper we give the first near-linear time approximation algorithm for this problem. Specifically speaking, we propose an $\tilde{O}(\frac{1}{\epsilon^7}m)$ time algorithm that computes a spanning tree with tree degree $(1+\epsilon)\Delta^* + O(\frac{1}{\epsilon^2}\log n)$ for any constant $\epsilon \in (0,\frac{1}{6})$. Thus, when $\Delta^*=\omega(\log n)$, we can achieve approximate solutions with constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page

Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

In this paper, we study the $k$-forest problem in the model of resource augmentation. In the $k$-forest problem, given an edge-weighted graph $G(V,E)$, a parameter $k$, and a set of $m$ demand pairs $\subseteq V \times V$, the objective is to construct a minimum-cost subgraph that connects at least $k$ demands. The problem is hard to approximate---the best-known approximation ratio is $O(\min\{\sqrt{n}, \sqrt{k}\})$. Furthermore, $k$-forest is as hard to approximate as the notoriously-hard densest $k$-subgraph problem. While the $k$-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are {\em not} connected. In particular, the objective of the $k$-forest problem can be viewed as to remove at most $m-k$ demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the $k$-forest problem that, for every $\epsilon>0$, removes at most $m-k$ demands and has cost no more than $O(1/\epsilon^{2})$ times the cost of an optimal algorithm that removes at most $(1-\epsilon)(m-k)$ demands

Algorithms for Cut Problems on Trees

We study the {\sc multicut on trees} and the {\sc generalized multiway Cut on trees} problems. For the {\sc multicut on trees} problem, we present a parameterized algorithm that runs in time $O^{*}(\rho^k)$, where $\rho = \sqrt{\sqrt{2} + 1} \approx 1.555$ is the positive root of the polynomial $x^4-2x^2-1$. This improves the current-best algorithm of Chen et al. that runs in time $O^{*}(1.619^k)$. For the {\sc generalized multiway cut on trees} problem, we show that this problem is solvable in polynomial time if the number of terminal sets is fixed; this answers an open question posed in a recent paper by Liu and Zhang. By reducing the {\sc generalized multiway cut on trees} problem to the {\sc multicut on trees} problem, our results give a parameterized algorithm that solves the {\sc generalized multiway cut on trees} problem in time $O^{*}(\rho^k)$, where $\rho = \sqrt{\sqrt{2} + 1} \approx 1.555$ time

Identifying subtypes of patients with neovascular age-related macular degeneration by genotypic and cardiovascular risk characteristics

<p>Abstract</p> <p>Background</p> <p>One of the challenges in the interpretation of studies showing associations between environmental and genotypic data with disease outcomes such as neovascular age-related macular degeneration (AMD) is understanding the phenotypic heterogeneity within a patient population with regard to any risk factor associated with the condition. This is critical when considering the potential therapeutic response of patients to any drug developed to treat the condition. In the present study, we identify patient subtypes or clusters which could represent several different targets for treatment development, based on genetic pathways in AMD and cardiovascular pathology.</p> <p>Methods</p> <p>We identified a sample of patients with neovascular AMD, that in previous studies had been shown to be at elevated risk for the disease through environmental factors such as cigarette smoking and genetic variants including the complement factor H gene (<it>CFH</it>) on chromosome 1q25 and variants in the <it>ARMS2</it>/HtrA serine peptidase 1 (<it>HTRA1</it>) gene(s) on chromosome 10q26. We conducted a multivariate segmentation analysis of 253 of these patients utilizing available epidemiologic and genetic data.</p> <p>Results</p> <p>In a multivariate model, cigarette smoking failed to differentiate subtypes of patients. However, four meaningfully distinct clusters of patients were identified that were most strongly differentiated by their cardiovascular health status (histories of hypercholesterolemia and hypertension), and the alleles of <it>ARMS2</it>/<it>HTRA1 </it>rs1049331.</p> <p>Conclusions</p> <p>These results have significant personalized medicine implications for drug developers attempting to determine the effective size of the treatable neovascular AMD population. Patient subtypes or clusters may represent different targets for therapeutic development based on genetic pathways in AMD and cardiovascular pathology, and treatments developed that may elevate CV risk, may be ill advised for certain of the clusters identified.</p

Brief Report: Whole-Exome Sequencing Revealing Somatic NLRP3 Mosaicism in a Patient With Chronic Infantile Neurologic, Cutaneous, Articular Syndrome.

To identify the genetic cause of chronic infantile neurologic, cutaneous, articular syndrome (CINCA syndrome) using whole-exome sequencing in a child who had typical clinical features but who was NLRP3 mutation negative based on conventional Sanger sequencing

Novel measures of cardiovascular health and its association with prevalence and progression of age-related macular degeneration: the CHARM study

<p>Abstract</p> <p>Background</p> <p>To determine if novel measures of cardiovascular health are associated with prevalence or progression of age-related macular degeneration (AMD).</p> <p>Methods</p> <p>Measures of the cardiovascular system: included intima media thickness (IMT), pulse wave velocity (PWV), systemic arterial compliance (SAC), carotid augmentation index (AI). For the prevalence study, hospital-based AMD cases and population-based age- and gender-matched controls with no signs of AMD in either eye were enrolled. For the progression component, participants with early AMD were recruited from two previous studies; cases were defined as progression in one or both eyes and controls were defined as no progression in either eye.</p> <p>Results</p> <p>160 cases and 160 controls were included in the prevalence component. The upper two quartiles of SAC, implying good cardiovascular health, were significantly associated with increased risk of AMD (OR = 2.54, 95% CL = 1.29, 4.99). High PWV was associated with increased prevalent AMD. Progression was observed in 82 (32.3%) of the 254 subjects recruited for the progression component. Higher AI (worse cardiovascular function) was protective for AMD progression (OR = 0.30, 95%CL = 0.13, 0.69). Higher aortic PWV was associated with increased risk of AMD progression; the highest risk was seen with the second lowest velocity (OR = 6.22, 95% CL = 2.35, 16.46).</p> <p>Conclusion</p> <p>The results were unexpected in that better cardiovascular health was associated with increased risk of prevalent AMD and progression. Inconsistent findings between the prevalence and progression components could be due to truly different disease etiologies or to spurious findings, as can occur with inherent biases in case control studies of prevalence. Further investigation of these non-invasive methods of characterizing the cardiovascular system should be undertaken as they may help to further elucidate the role of the cardiovascular system in the etiology of prevalent AMD and progression.</p

Proof of concept, randomized, placebo-controlled study of the effect of simvastatin on the course of age-related macular degeneration

BACKGROUND: HMG Co-A reductase inhibitors are ubiquitous in our community yet their potential role in age-related macular degeneration (AMD) remains to be determined. METHODOLOGY/PRINCIPAL FINDINGS: OBJECTIVES: To evaluate the effect of simvastatin on AMD progression and the effect modification by polymorphism in apolipoprotein E (ApoE) and complement factor H (CFH) genes. DESIGN: A proof of concept double-masked randomized controlled study. PARTICIPANTS: 114 participants aged 53 to 91 years, with either bilateral intermediate AMD or unilateral non-advanced AMD (with advanced AMD in fellow eye), BCVA ≥ 20/60 in at least one eye, and a normal lipid profile. INTERVENTION: Simvastatin 40 mg/day or placebo, allocated 1:1. MAIN OUTCOME MEASURES: Progression of AMD either to advanced AMD or in severity of non-advanced AMD. Results. The cumulative AMD progression rates were 70% in the placebo and 54% in the simvastatin group. Intent to treat multivariable logistic regression analysis, adjusted for age, sex, smoking and baseline AMD severity, showed a significant 2-fold decrease in the risk of progression in the simvastatin group: OR 0.43 (0.18-0.99), p = 0.047. Post-hoc analysis stratified by baseline AMD severity showed no benefit from treatment in those who had advanced AMD in the fellow eye before enrolment: OR 0.97 (0.27-3.52), p = 0.96, after adjusting for age, sex and smoking. However, there was a significant reduction in the risk of progression in the bilateral intermediate AMD group compared to placebo [adjusted OR 0.23 (0.07-0.75), p = 0.015]. The most prominent effect was observed amongst those who had the CC (Y402H) at risk genotype of the CFH gene [OR 0.08 (0.02-0.45), p = 0.004]. No evidence of harm from simvastatin intervention was detected. CONCLUSION/SIGNIFICANCE: Simvastatin may slow progression of non-advanced AMD, especially for those with the at risk CFH genotype CC (Y402H). Further exploration of the potential use of statins for AMD, with emphasis on genetic subgroups, is warranted. TRIAL REGISTRATION: Australian New Zealand Clinical Trial Registry (ANZCTR) ACTRN1260500032065