Satyam Singh

Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh et al.

We give an approximation scheme for the TSP in $d$-dimensional hyperbolic space that has optimal dependence on $\varepsilon$ under Gap-ETH. For any fixed dimension $d\geq 2$ and for any $\varepsilon>0$ our randomized algorithm gives a $(1+\varepsilon)$-approximation in time $2^{O(1/\varepsilon^{d-1})}n^{1+o(1)}$. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.

Minati De, Satyam Singh

In the classical online model, the maximum independent set problem admits an $Ω(n)$ lower bound on the competitive ratio even for interval graphs, motivating the study of the problem under additional assumptions. We first study the problem on graphs with a bounded independent kissing number $ζ$, defined as the size of the largest induced star in the graph minus one. We show that a simple greedy algorithm, requiring no geometric representation, achieves a competitive ratio of $ζ$. Moreover, this bound is optimal for deterministic online algorithms and asymptotically optimal for randomized ones. This extends previous results from specific geometric graph families to more general graph classes. Since this bound rules out further improvements through randomization alone, we investigate the power of randomization with access to geometric representation. When the geometric representation of the objects is known, we present randomized online algorithms with improved guarantees. For unit ball graphs in $\mathbb{R}^3$, we present an algorithm whose expected competitive ratio is strictly smaller than the deterministic lower bound implied by the independent kissing number. For $α$-fat objects and for axis-aligned hyper-rectangles in $\mathbb{R}^d$ with bounded diameters, we obtain algorithms with expected competitive ratios that depend polylogarithmically on the ratio between the maximum and minimum object diameters. In both cases, the randomized lower bound implied by the independent kissing number grows polynomially with the ratio between the maximum and minimum object diameters, implying substantial performance guarantees for our algorithms.

Ashok Mondal, Satyam Singh

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.