Improvements on the k-center problem for uncertain data

In real applications, there are situations where we need to model some problems based on uncertain data. This leads us to define an uncertain model for some classical geometric optimization problems and propose algorithms to solve them. In this paper, we study the $k$-center problem, for uncertain input. In our setting, each uncertain point $P_i$ is located independently from other points in one of several possible locations $\{P_{i,1},\dots, P_{i,z_i}\}$ in a metric space with metric $d$, with specified probabilities and the goal is to compute $k$-centers $\{c_1,\dots, c_k\}$ that minimize the following expected cost $$Ecost(c_1,\dots, c_k)=\sum_{R\in \Omega} prob(R)\max_{i=1,\dots, n}\min_{j=1,\dots k} d(\hat{P}_i,c_j)$$ here $\Omega$ is the probability space of all realizations $$R=\{\hat{P}_1,\dots, \hat{P}_n\}$$ of given uncertain points and $$prob(R)=\prod_{i=1}^n prob(\hat{P}_i).$$ In restricted assigned version of this problem, an assignment $A:\{P_1,\dots, P_n\}\rightarrow \{c_1,\dots, c_k\}$ is given for any choice of centers and the goal is to minimize $$Ecost_A(c_1,\dots, c_k)=\sum_{R\in \Omega} prob(R)\max_{i=1,\dots, n} d(\hat{P}_i,A(P_i)).$$ In unrestricted version, the assignment is not specified and the goal is to compute $k$ centers $\{c_1,\dots, c_k\}$ and an assignment $A$ that minimize the above expected cost. We give several improved constant approximation factor algorithms for the assigned versions of this problem in a Euclidean space and in a general metric space. Our results significantly improve the results of \cite{guh} and generalize the results of \cite{wang} to any dimension. Our approach is to replace a certain center point for each uncertain point and study the properties of these certain points. The proposed algorithms are efficient and simple to implement

Brief Announcement: Distributed Algorithms for Minimum Dominating Set Problem and Beyond, a New Approach

In this paper, we study the minimum dominating set (MDS) problem and the minimum total dominating set (MTDS) problem. We propose a new idea to compute approximate MDS and MTDS. This new approach can be implemented in a distributed model or parallel model. We also show how to use this new approach in other related problems such as set cover problem and k-distance dominating set problem

Relative Fractional Independence Number and Its Applications

We define the relative fractional independence number of two graphs, $G$ and $H$, as $$\alpha^*(G|H)=\max_{W}\frac{\alpha(G\boxtimes W)}{\alpha(H\boxtimes W)},$$ where the maximum is taken over all graphs $W$, $G\boxtimes W$ is the strong product of $G$ and $W$, and $\alpha$ denotes the independence number. We give a non-trivial linear program to compute $\alpha^*(G|H)$ and discuss some of its properties. We show that $$\alpha^*(G|H)\geq \frac{X(G)}{X(H)},$$ where $X(G)$ can be the independence number, the zero-error Shannon capacity, the fractional independence number, the Lov'{a}sz number, or the Schrijver's or Szegedy's variants of the Lov'{a}sz number of a graph $G$. This inequality is the first explicit non-trivial upper bound on the ratio of the invariants of two arbitrary graphs, as mentioned earlier, which can also be used to obtain upper or lower bounds for these invariants. As explicit applications, we present new upper bounds for the ratio of the zero-error Shannon capacity of two Cayley graphs and compute new lower bounds on the Shannon capacity of certain Johnson graphs (yielding the exact value of their Haemers number). Moreover, we show that the relative fractional independence number can be used to present a stronger version of the well-known No-Homomorphism Lemma. The No-Homomorphism Lemma is widely used to show the non-existence of a homomorphism between two graphs and is also used to give an upper bound on the independence number of a graph. Our extension of the No-Homomorphism Lemma is computationally more accessible than its original version

Estimates, trends, and drivers of the global burden of type 2 diabetes attributable to PM<inf>2·5</inf> air pollution, 1990–2019: an analysis of data from the Global Burden of Disease Study 2019

Background: Experimental and epidemiological studies indicate an association between exposure to particulate matter (PM) air pollution and increased risk of type 2 diabetes. In view of the high and increasing prevalence of diabetes, we aimed to quantify the burden of type 2 diabetes attributable to PM2·5 originating from ambient and household air pollution. Methods: We systematically compiled all relevant cohort and case-control studies assessing the effect of exposure to household and ambient fine particulate matter (PM2·5) air pollution on type 2 diabetes incidence and mortality. We derived an exposure–response curve from the extracted relative risk estimates using the MR-BRT (meta-regression—Bayesian, regularised, trimmed) tool. The estimated curve was linked to ambient and household PM2·5 exposures from the Global Burden of Diseases, Injuries, and Risk Factors Study 2019, and estimates of the attributable burden (population attributable fractions and rates per 100 000 population of deaths and disability-adjusted life-years) for 204 countries from 1990 to 2019 were calculated. We also assessed the role of changes in exposure, population size, age, and type 2 diabetes incidence in the observed trend in PM2·5-attributable type 2 diabetes burden. All estimates are presented with 95% uncertainty intervals. Findings: In 2019, approximately a fifth of the global burden of type 2 diabetes was attributable to PM2·5 exposure, with an estimated 3·78 (95% uncertainty interval 2·68–4·83) deaths per 100 000 population and 167 (117–223) disability-adjusted life-years (DALYs) per 100 000 population. Approximately 13·4% (9·49–17·5) of deaths and 13·6% (9·73–17·9) of DALYs due to type 2 diabetes were contributed by ambient PM2·5, and 6·50% (4·22–9·53) of deaths and 5·92% (3·81–8·64) of DALYs by household air pollution. High burdens, in terms of numbers as well as rates, were estimated in Asia, sub-Saharan Africa, and South America. Since 1990, the attributable burden has increased by 50%, driven largely by population growth and ageing. Globally, the impact of reductions in household air pollution was largely offset by increased ambient PM2·5. Interpretation: Air pollution is a major risk factor for diabetes. We estimated that about a fifth of the global burden of type 2 diabetes is attributable PM2·5 pollution. Air pollution mitigation therefore might have an essential role in reducing the global disease burden resulting from type 2 diabetes. Funding: Bill & Melinda Gates Foundation