Improved Approximation Algorithm for Set Multicover with Non-Piercing Regions

In the Set Multicover problem, we are given a set system (X,?), where X is a finite ground set, and ? is a collection of subsets of X. Each element x ? X has a non-negative demand d(x). The goal is to pick a smallest cardinality sub-collection ?\u27 of ? such that each point is covered by at least d(x) sets from ?\u27. In this paper, we study the set multicover problem for set systems defined by points and non-piercing regions in the plane, which includes disks, pseudodisks, k-admissible regions, squares, unit height rectangles, homothets of convex sets, upward paths on a tree, etc. We give a polynomial time (2+?)-approximation algorithm for the set multicover problem (P, ?), where P is a set of points with demands, and ? is a set of non-piercing regions, as well as for the set multicover problem (?, P), where ? is a set of pseudodisks with demands, and P is a set of points in the plane, which is the hitting set problem with demands

On Hypergraph Supports

Let $\mathcal{H}=(X,\mathcal{E})$ be a hypergraph. A support is a graph $Q$ on $X$ such that for each $E\in\mathcal{E}$, the subgraph of $Q$ induced on the elements in $E$ is connected. In this paper, we consider hypergraphs defined on a host graph. Given a graph $G=(V,E)$, with $c:V\to\{\mathbf{r},\mathbf{b}\}$, and a collection of connected subgraphs $\mathcal{H}$ of $G$, a primal support is a graph $Q$ on $\mathbf{b}(V)$ such that for each $H\in \mathcal{H}$, the induced subgraph $Q[\mathbf{b}(H)]$ on vertices $\mathbf{b}(H)=H\cap c^{-1}(\mathbf{b})$ is connected. A \emph{dual support} is a graph $Q^*$ on $\mathcal{H}$ s.t. for each $v\in X$, the induced subgraph $Q^*[\mathcal{H}_v]$ is connected, where $\mathcal{H}_v=\{H\in\mathcal{H}: v\in H\}$. We present sufficient conditions on the host graph and hyperedges so that the resulting support comes from a restricted family. We primarily study two classes of graphs: $(1)$ If the host graph has genus $g$ and the hypergraphs satisfy a topological condition of being \emph{cross-free}, then there is a primal and a dual support of genus at most $g$. $(2)$ If the host graph has treewidth $t$ and the hyperedges satisfy a combinatorial condition of being \emph{non-piercing}, then there exist primal and dual supports of treewidth $O(2^t)$. We show that this exponential blow-up is sometimes necessary. As an intermediate case, we also study the case when the host graph is outerplanar. Finally, we show applications of our results to packing and covering, and coloring problems on geometric hypergraphs

Planar Support for Non-piercing Regions and Applications

Given a hypergraph H=(X,S), a planar support for H is a planar graph G with vertex set X, such that for each hyperedge S in S, the sub-graph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization. The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph H_R(B) = (B, {B_r}_{r in R}), where B_r = {b in B: b cap r != empty set} has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in R cup B. Special cases of this result include the setting where either the family R, or the family B is a set of points. Our result unifies and generalizes several previous results on planar supports, PTASs for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions

QPTAS for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

International audienceWeighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal and Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the (1 + status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al. (SODA 2012), yielded an O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes half-spaces, disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in R 3 , for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP DTIME(2 polylog(n)). Together with the recent work of Chan and Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems

On Geometric Priority Set Cover Problems

We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane

Asymmetric severity of diabetic retinopathy in Waardenburg syndrome: response to authors

Aditi Gupta, Rajiv Raman, Tarun SharmaShri Bhagwan Mahavir Department of Vitreoretinal Services, Sankara Nethralaya, Chennai, Tamil Nadu, IndiaWe read with great interest the recent article by Kashima et al,1 in which the authors report a case of asymmetric severity of diabetic retinopathy in Waardenburg syndrome. We want to highlight some concerns regarding this report. Previous reports have described many systemic and local factors associated with the development of asymmetric diabetic retinopathy.2,3 These include myopia ≥5 D, anisometropia >1 D, amblyopia, unilateral elevated intraocular pressure, complete posterior vitreous detachment, unilateral carotid artery stenosis, ocular ischemic syndrome, and chorioretinal scarring.2,3 In any suspected case of asymmetric diabetic retinopathy, it is prudent to rule out the abovementioned factors first. In the present case, although the authors clearly mention the absence of internal carotid and ophthalmic artery obstruction on magnetic resonance angiography, it would have been more informative if the authors had also provided the refractive error, intraocular pressure, and posterior vitreous detachment status of both the eyes.Likewise, it would have been useful to note the arm-retina time and retinal arteriovenous filling time in both the eyes on fundus fluorescein angiography, which is usually used to diagnose ocular ischemic syndrome by monitoring extension of the retinal circulation time, including time of blood circulation from the arm to the retina and the retinal arteriovenous filling time.4,5 The mere absence of internal carotid obstruction on magnetic resonance angiography cannot rule out the presence of ocular ischemic syndrome because, rarely, ocular ischemic syndrome can also occur secondary to other causes, such as arteritis.6,7 Comparing the arm-retina time and retinal arteriovenous filling time on fundus fluorescein angiography in both the eyes would be more helpful to rule out ocular ischemic syndrome

Traumatic Cyclodialysis Cleft Surgical Management Using Encirclage and Cryotherapy : A Novel Approach

This is a retrospective case series of 4 patients with traumatic cyclodialysis cleft (CC) with features of hypotony and posterior segment manifestations of blunt trauma who were treated using encirclage and trans scleral cryoptherapy along with vitrectomy. Encirclage was placed anteriorly in order to support the cleft. There was closure of cleft and improvement in Intra ocular pressure (IOP) in all cases. Thus the identification and treatment of CC with encirclage in cases with varied posterior manifestation of trauma can lead to good anatomical and visual restoration

Packing and Covering with Non-Piercing Regions

In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks). We show that the local search algorithm that yields PTASs when the regions are disks [Aschner/Katz/Morgenstern/Yuditsky, WALCOM 2013; Gibson/Pirwani, 2005; Mustafa/Raman/Ray, 2015] can be extended to work for non-piercing regions. While such an extension is intuitive and natural, attempts to settle this question have failed even for pseudodisks. The techniques used for analysis when the regions are disks rely heavily on the underlying geometry, and do not extend to topologically defined settings such as pseudodisks. In order to prove our results, we introduce novel techniques that we believe will find applications in other problems. We then consider the Capacitated Region Packing problem. Here, the input consists of a set of points with capacities, and a set of regions. The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity. We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded. Our result settles a conjecture of Har-Peled (see Conclusion of [Har-Peled, SoCG 2014]) in the affirmative. The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane. Finally, we consider the Capacitated Point Packing problem. In this setting, the regions have capacities, and our objective is to find a maximum cardinality subset of points such that no region has more points than its capacity. We show that this problem admits a PTAS when the capacity is unity, extending one of the results of Ene et al. [Ene/Har-Peled/Raichel, SoCG 2012]

Revised Glycemic Index for Diagnosing and Monitoring of Diabetes Mellitus in South Indian Population

AIM: To find the optimal threshold of fasting plasma glucose (FPG) and glycated hemoglobin (HbA1c) for diagnosis of diabetes mellitus (DM) and to evaluate the association with diabetic retinopathy (DR) in the South Indian population. SETTINGS AND DESIGN: A retrospective population-based study. METHODS AND MATERIALS: A total of 909 newly detected type 2 DM patients were selected from our two previously conducted studies, which include an urban and a rural population of South India. All underwent estimation of fasting, postprandial plasma glucose (PPG), and other biochemical tests. A comprehensive and detailed ophthalmic examination was carried out. The fundi of patients were photographed using 45°, four-field stereoscopic photography. Based on receiver operating characteristic (ROC) curves, sensitivity and specificity were derived. RESULTS:  The optimal cut-off values determined by maximizing the sensitivity and specificity of FPG and HbA1c using the Youden index were ≥ 6.17 mmol/L and ≥ 6.3%, respectively. By distributing the cut-off points into deciles and comparing them to the WHO criteria, we found that our HbA1c level of 6.60% was more than the WHO threshold (6.5%), with higher sensitivity (81.6%) and lower specificity (48.3%). The FPG level of 6.80 mmol/L was lower to the WHO criteria (7 mmol/L) with increased sensitivity (77.0%) and lower specificity (45.7%). Prevalence of DR by HbA1c levels between 6.5% and 6.9% was 15.3%. The prevalence of DR was more in the FPG category between 6.4 and 6.9 mmol/L and ≥ 7.5 mmol/L. CONCLUSION: Our population-based data indicate that for the South Indian population HbA1c value of ≥63 % and FPG value of ≥6.17 mmol/L may be optimal for diagnosing DM with a high level of accuracy and will be useful for the identification of mild and moderate DR

Need to improve awareness and treatment compliance in high-risk patients for diabetic complications in Nepal

Objective/introduction It is known that knowledge, awareness, and practice influence diabetic control. We compared factors pertaining to healthy lifestyle (exercising, avoiding smoking), self-help (attending appointments, following treatment regimens), and diabetic awareness in high-risk patients for diabetic complications, specifically, those on insulin versus non-insulin treatment, and also those with a longer diabetic duration (≥5 years) versus a shorter duration. Methods 200 consecutive patients with type 2 diabetes (52.0±11.6 years) attending diabetic clinic at a referral hospital in Nepal were recruited. A structured questionnaire explored non-clinical parameters including age, gender, diabetic duration, awareness about diabetes control, self-help, and lifestyle. Clinical data were also measured: HbA1c, fasting blood sugar (FBS), blood pressure, and treatment type (insulin, diet/tablet). Results A significantly higher proportion of patients on insulin (vs non-insulin) or with diabetic duration ≥5 years (vs <5 years) self-reported not doing regular exercise, forgetting to take medicine, and not knowing whether their diabetes was controlled (p≤0.005). HbA1c/FBS levels were significantly higher for patients on insulin or with a longer diabetic duration (p≤0.001). 92% of those on insulin (vs 31% on non-insulin) and 91% with diabetic duration ≥5 years (vs 28% of <5 years) self-reported to seeking medical help due to episodes of uncontrolled blood sugar in the last year (p<0.001). Conclusion Poor self-help/lifestyle and reduced knowledge/awareness about diabetic control was found in patients on insulin or with longer diabetic duration. This is a worrying finding as these patients are already at high risk for developing diabetic complications. The findings highlight need for targeting this more vulnerable group and provide more support/diabetic educational tools