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

Use of multidimensional item response theory methods for dementia prevalence prediction : an example using the Health and Retirement Survey and the Aging, Demographics, and Memory Study

Background Data sparsity is a major limitation to estimating national and global dementia burden. Surveys with full diagnostic evaluations of dementia prevalence are prohibitively resource-intensive in many settings. However, validation samples from nationally representative surveys allow for the development of algorithms for the prediction of dementia prevalence nationally. Methods Using cognitive testing data and data on functional limitations from Wave A (2001-2003) of the ADAMS study (n = 744) and the 2000 wave of the HRS study (n = 6358) we estimated a two-dimensional item response theory model to calculate cognition and function scores for all individuals over 70. Based on diagnostic information from the formal clinical adjudication in ADAMS, we fit a logistic regression model for the classification of dementia status using cognition and function scores and applied this algorithm to the full HRS sample to calculate dementia prevalence by age and sex. Results Our algorithm had a cross-validated predictive accuracy of 88% (86-90), and an area under the curve of 0.97 (0.97-0.98) in ADAMS. Prevalence was higher in females than males and increased over age, with a prevalence of 4% (3-4) in individuals 70-79, 11% (9-12) in individuals 80-89 years old, and 28% (22-35) in those 90 and older. Conclusions Our model had similar or better accuracy as compared to previously reviewed algorithms for the prediction of dementia prevalence in HRS, while utilizing more flexible methods. These methods could be more easily generalized and utilized to estimate dementia prevalence in other national surveys