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VC-dimension and $\varepsilon$-nets are key concepts in Statistical Learning Theory. Intuitively, VC-dimension is a measure of the size of a class of sets. The famous $\varepsilon$-net theorem, a fundamental result in Discrete Geometry, asserts that if the VC-dimension of a set system is bounded, then a small sample exists that intersects all sufficiently large sets. In online learning scenarios where data arrives sequentially, the VC-dimension helps to bound the complexity of the set system, and $\varepsilon$-nets ensure the selection of a small representative set. This sampling framework is crucial in various domains, including spatial data analysis, motion planning in dynamic environments, optimization of sensor networks, and feature extraction in computer vision, among others. Motivated by these applications, we study the online $\varepsilon$-net problem for geometric concepts with bounded VC-dimension. While the offline version of this problem has been extensively studied, surprisingly, there are no known theoretical results for the online version to date. We present the first deterministic online algorithm with an optimal competitive ratio for intervals in $\mathbb{R}$. Next, we give a randomized online algorithm with a near-optimal competitive ratio for axis-aligned boxes in $\mathbb{R}^d$, for $d\le 3$. Furthermore, we introduce a novel technique to analyze similar-sized objects of constant description complexity in $\mathbb{R}^d$, which may be of independent interest. Next, we focus on the continuous version of this problem, where ranges of the set system are geometric concepts in $\mathbb{R}^d$ arriving in an online manner, but the universe is the entire space, and the objective is to choose a small sample that intersects all the ranges., Comment: 18 pages, 4 Figures

The hitting set problem is one of the fundamental problems in combinatorial optimization and is well-studied in offline setup. We consider the online hitting set problem, where only the set of points is known in advance, and objects are introduced one by one. Our objective is to maintain a minimum-sized hitting set by making irrevocable decisions. Here, we present the study of two variants of the online hitting set problem depending on the point set. In the first variant, we consider the point set to be the entire $\mathbb{Z}^d$, while in the second variant, we consider the point set to be a finite subset of $\mathbb{R}^2$. If you use points in $\mathbb{Z}^d$ to hit homothetic hypercubes in $\mathbb{R}^d$ with side lengths in $[1,M]$, we show that the competitive ratio of any algorithm is $\Omega(d\log M)$, whether it is deterministic or random. This improves the recently known deterministic lower bound of $\Omega(\log M)$ by a factor of $d$. Then, we present an almost tight randomized algorithm with a competitive ratio $O(d^2\log M)$ that significantly improves the best-known competitive ratio of $25^d\log M$. Next, we propose a simple deterministic ${\lfloor\frac{2}{\alpha}+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ competitive algorithm to hit similarly sized {$\alpha$-fat objects} in $\mathbb{R}^d$ having diameters in the range $[1, M]$ using points in $\mathbb{Z}^d$. This improves the current best-known upper bound by a factor of at least $5^d$. Finally, we consider the hitting set problem when the point set consists of $n$ points in $\mathbb{R}^2$, and the objects are homothetic regular $k$-gons having diameter in the range $[1, M]$. We present an $O(\log n\log M)$ competitive randomized algorithm for that. Whereas no result was known even for squares. In particular, our results answer some of the open questions raised by Khan et al. (SoCG'23) and Alefkhani et al. (WAOA'23)., Comment: 30 pages and 10 figures. arXiv admin note: text overlap with arXiv:2304.06780

In recent years, large-scale adoption of cloud storage solutions has revolutionized the way we think about digital data storage. However, the exponential increase in data volume, especially images, has raised environmental concerns regarding power and resource consumption, as well as the rising digital carbon footprint emissions. The aim of this research is to propose a methodology for cloud-based image storage by integrating image compression technology with SuperResolution Generative Adversarial Networks (SRGAN). Rather than storing images in their original format directly on the cloud, our approach involves initially reducing the image size through compression and downsizing techniques before storage. Upon request, these compressed images will be retrieved and processed by SRGAN to generate images. The efficacy of the proposed method is evaluated in terms of PSNR and SSIM metrics. Additionally, a mathematical analysis is given to calculate power consumption and carbon footprint assesment. The proposed data compression technique provides a significant solution to achieve a reasonable trade off between environmental sustainability and industrial efficiency., Comment: 5 pages, 5 figures

In today's technology-driven era, the imperative for predictive maintenance and advanced diagnostics extends beyond aviation to encompass the identification of damages, failures, and operational defects in rotating and moving machines. Implementing such services not only curtails maintenance costs but also extends machine lifespan, ensuring heightened operational efficiency. Moreover, it serves as a preventive measure against potential accidents or catastrophic events. The advent of Artificial Intelligence (AI) has revolutionized maintenance across industries, enabling more accurate and efficient prediction and analysis of machine failures, thereby conserving time and resources. Our proposed study aims to delve into various machine learning classification techniques, including Support Vector Machine (SVM), Random Forest, Logistic Regression, and Convolutional Neural Network LSTM-Based, for predicting and analyzing machine performance. SVM classifies data into different categories based on their positions in a multidimensional space, while Random Forest employs ensemble learning to create multiple decision trees for classification. Logistic Regression predicts the probability of binary outcomes using input data. The primary objective of the study is to assess these algorithms' performance in predicting and analyzing machine performance, considering factors such as accuracy, precision, recall, and F1 score. The findings will aid maintenance experts in selecting the most suitable machine learning algorithm for effective prediction and analysis of machine performance.

We present online deterministic algorithms for minimum coloring and minimum dominating set problems in the context of geometric intersection graphs. We consider a graph parameter: the independent kissing number $\zeta$, which is a number equal to `the size of the largest induced star in the graph $-1$'. For a graph with an independent kissing number at most $\zeta$, we show that the famous greedy algorithm achieves an optimal competitive ratio of $\zeta$ for the minimum dominating set and the minimum independent dominating set problems. However, for the minimum connected dominating set problem, we obtain a competitive ratio of at most $2\zeta$. To complement this, we prove that for the minimum connected dominating set problem, any deterministic online algorithm has a competitive ratio of at least $2(\zeta-1)$ for the geometric intersection graph of translates of a convex object in $\mathbb{R}^2$. Next, for the minimum coloring problem, we obtain algorithms having a competitive ratio of $O\left({\zeta'}{\log m}\right)$ for geometric intersection graphs of bounded scaled $\alpha$-fat objects in $\mathbb{R}^d$ having widths in the interval $[1,m]$, where $\zeta'$ is the independent kissing number of the geometric intersection graph of bounded scaled $\alpha$-fat objects having widths in the interval $[1,2]$. Finally, we investigate the value of $\zeta$ for geometric intersection graphs of various families of geometric objects., Comment: 20 pages, 7 figures and 1 table. arXiv admin note: text overlap with arXiv:2111.07812

We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic algorithm solving this problem for intervals in $\mathbb{R}$ has a competitive ratio of at least $\Omega(n)$. This paper considers the piercing set problem for similarly sized objects. We propose a deterministic online algorithm for similarly sized fat objects in $\mathbb{R}^d$. For homothetic hypercubes in $\mathbb{R}^d$ with side length in the range $[1,k]$, we propose a deterministic algorithm having a competitive ratio of at most~$3^d\lceil\log_2 k\rceil+2^d$. In the end, we show deterministic lower bounds of the competitive ratio for similarly sized $\alpha$-fat objects in $\mathbb{R}^2$ and homothetic hypercubes in $\mathbb{R}^d$. Note that piercing translated copies of a convex object is equivalent to the unit covering problem, which is well-studied in the online setup. Surprisingly, no upper bound of the competitive ratio was known for the unit covering problem when the corresponding object is anything other than a ball or a hypercube. Our result yields an upper bound of the competitive ratio for the unit covering problem when the corresponding object is any convex object in $\mathbb{R}^d$., Comment: 21 pages and 9 figures

We investigate the geometric hitting set problem in the online setup for the range space $\Sigma=({\cal P},{\cal S})$, where the set $\P\subset\mathbb{R}^2$ is a collection of $n$ points and the set $\cal S$ is a family of geometric objects in $\mathbb{R}^2$. In the online setting, the geometric objects arrive one by one. Upon the arrival of an object, an online algorithm must maintain a valid hitting set by making an irreversible decision, i.e., once a point is added to the hitting set by the algorithm, it can not be deleted in the future. The objective of the geometric hitting set problem is to find a hitting set of the minimum cardinality. Even and Smorodinsky (Discret. Appl. Math., 2014) considered an online model (Model-I) in which the range space $\Sigma$ is known in advance, but the order of arrival of the input objects in $\cal S$ is unknown. They proposed online algorithms having optimal competitive ratios of $\Theta(\log n)$ for intervals, half-planes and unit disks in $\mathbb{R}^2$. Whether such an algorithm exists for unit squares remained open for a long time. This paper considers an online model (Model-II) in which the entire range space $\Sigma$ is not known in advance. We only know the set $\cal P$ but not the set $\cal S$ in advance. Note that any algorithm for Model-II will also work for Model-I, but not vice-versa. In Model-II, we obtain an optimal competitive ratio of $\Theta(\log(n))$ for unit disks and regular $k$-gon with $k\geq 4$ in $\mathbb{R}^2$. All the above-mentioned results also hold for the equivalent geometric set cover problem in Model-II., Comment: 13 pages, 9 figures

We consider the online hitting set problem for the range space $\Sigma=(\cal X,\cal R)$, where the point set $\cal X$ is known beforehand, but the set $\cal R$ of geometric objects is not known in advance. Here, objects from $\cal R$ arrive one by one. The objective of the problem is to maintain a hitting set of the minimum cardinality by taking irrevocable decisions. In this paper, we consider the problem when objects are unit balls or unit hypercubes in $\mathbb{R}^d$, and the points from $\mathbb{Z}^d$ are used for hitting them. First, we address the case when objects are unit intervals in $\mathbb{R}$ and present an optimal deterministic algorithm with a competitive ratio of~$2$. Then, we consider the case when objects are unit balls. For hitting unit balls in $\mathbb{R}^2$ and $\mathbb{R}^3$, we present $4$ and $14$-competitive deterministic algorithms, respectively. On the other hand, for hitting unit balls in $\mathbb{R}^d$, we propose an $O(d^4)$-competitive deterministic algorithm, and we demonstrate that}, for $d<4$, the competitive ratio of any deterministic algorithm is at least $d+1$. In the end, we explore the case where objects are unit hypercubes. For hitting unit hypercubes in $\mathbb{R}^2$ and $\mathbb{R}^3$, we obtain $4$ and $8$-competitive deterministic algorithms, respectively. For hitting unit hypercubes in $\mathbb{R}^d$ ($d\geq 3$), we present an $O(d^2)$-competitive randomized algorithm. Furthermore, we prove that the competitive ratio of any deterministic algorithm for the problem is at least $d+1$ for any $d\in\mathbb{N}$., Comment: 20 pages, 5 figures. Accepted in TCS

Finding minimum dominating set and maximum independent set for graphs in the classical online setup are notorious due to their disastrous $\Omega(n)$ lower bound of the competitive ratio that even holds for interval graphs, where $n$ is the number of vertices. In this paper, inspired by Newton number, first, we introduce the independent kissing number $\zeta$ of a graph. We prove that the well known online greedy algorithm for dominating set achieves optimal competitive ratio $\zeta$ for any graph. We show that the same greedy algorithm achieves optimal competitive ratio $\zeta$ for online maximum independent set of a class of graphs with independent kissing number $\zeta$. For minimum connected dominating set problem, we prove that online greedy algorithm achieves an asymptotic competitive ratio of $2(\zeta-1)$, whereas for a family of translated convex objects the lower bound is $\frac{2\zeta-1}{3}$. Finally, we study the value of $\zeta$ for some specific families of geometric objects: fixed and arbitrary oriented unit hyper-cubes in $I\!\!R^d$, congruent balls in $I\!\!R^3$, fixed oriented unit triangles, fixed and arbitrary oriented regular polygons in $I\!\!R^2$. For each of these families, we also present lower bounds of the minimum connected dominating set problem., Comment: 26 pages, 17 figures