arXiv subjects

Rajiv Raman

Publications and source records attributed to Rajiv Raman.

15 recordsLinked to original sources

arXiv2026

Counterexamples to Wegner's Conjecture for Rectangles

Wegner conjectured in 1965 that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $\tau(\mathcal R)\le 2\nu(\mathcal R)-1$, where $\tau(\mathcal R)$ is the minimum number of piercing points and $\nu(\mathcal R)$ is the maximum size of a pairwise-disjoint subfamily. We disprove the conjecture by an explicit triangle-free family of $64$ rectangles with $\nu=16$ and $\tau\ge 32$. More generally, for every $\varepsilon>0$, we construct triangle-free rectangle families for which the standard clique-LP relaxation for maximum independent set of rectangles has integrality gap at least $5/2-\varepsilon$. The same families satisfy $\tau(\mathcal R)\ge (5/2-\varepsilon)\nu(\mathcal R)$. We also prove that, on triangle-free rectangle families, this LP has gap at most $3$. Our approach gives an example with axis-parallel segments instead of rectangles with integrality gap tending to $2$. We also give a relatively small $4092$-rectangle triangle-free family with chromatic number $6$ improving the construction of Asplund and Gr\"unbaum (On a coloring problem, Mathematica Scandinavica, 1960) that required more than $10^8$ rectangles.

cs.CG↗

arXiv2026

Hitting Axis-Parallel Segments with Weighted Points

We study a geometric hitting-set problem in which the input consists of a set $P$ of weighted points and a family $S=H\cup V$ of axis-parallel segments in the plane. The goal is to select a minimum-weight subset of $P$ that hits every segment in $S$. Even restricted geometric hitting-set problems are known to be computationally hard, and for axis-parallel segments the standard decomposition into horizontal and vertical sub-instances yields only a simple factor-$2$ approximation. We present an LP-rounding algorithm that breaks the factor-2 barrier. For the weighted problem, we obtain a randomized $(1+2/e)$-approximation by combining systematic rounding on horizontal lines with an exact repair step on residual vertical sub-instances. In the unweighted case, a sharper analysis gives a $(1+1/(e-1))$-approximation. Finally, we consider the case where one of the sub-instances consists of lines instead of line segments, a problem considered by Fekete et al. (Geometric Hitting Set for Segments of Few Orientations, Theor. Comp. Sys., 62 (2) 2018),. In this case, we improve their result to obtain an approximation factor of $1+1/e$ and show that the problem is APX-hard. We also present algorithms for the generalization to $d$ orientations, as well as PTASes for bounded-complexity subclasses of the unweighted Hitting Set problem.

cs.CG↗

arXiv2026

Managing Diabetic Retinopathy with Deep Learning: A Data Centric Overview

Diabetic Retinopathy (DR) is a serious microvascular complication of diabetes, and one of the leading causes of vision loss worldwide. Although automated detection and grading, with Deep Learning (DL), can reduce the burden on ophthalmologists, it is constrained by the limited availability of high-quality datasets. Existing repositories often remain geographically narrow, contain limited samples, and exhibit inconsistent annotations or variable image quality; thereby, restricting their clinical reliability. This paper presents a comprehensive review and comparative analysis of fundus image datasets used in the management of DR. The study evaluates their usability across key tasks, including binary classification, severity grading, lesion localization, and multi-disease screening. It also categorizes the datasets by size, accessibility, and annotation type (such as image-level, lesion-level, and multi-disease). Finally, a recently published dataset is presented as a case study to illustrate broader challenges in dataset curation and usage. The review consolidates current knowledge while highlighting persistent gaps such as the lack of standardized lesion-level annotations and longitudinal data. It also outlines recommendations for future dataset development to support clinically reliable and explainable solutions in DR screening.

eess.IV↗

arXiv2026

Sweeping Arrangements of Non-Piercing Curves in Plane

Let $Γ$ be an arrangement of Jordan curves in the plane, i.e., simple closed curves in the plane. For any curve $γ\in Γ$, we denote the bounded region enclosed by $γ$ as $\tildeγ$. We say that $Γ$ is non-piercing if for any two curves $α, β\in Γ$, $\tildeα \,\setminus\, \tildeβ$ is connected. A non-piercing arrangement of curves generalizes a set of $2$-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG '89) proved that if we are given an arrangement $Γ$ of $2$-intersecting curves and a {\em sweep} curve $γ\inΓ$, then the arrangement can be \emph{swept} by $γ$ while always maintaining the $2$-intersecting property of the curves in $Γ$. We generalize the result of Snoeyink and Hershberger to the setting of non-piercing arrangements. Given an arrangement $Γ$ of non-piercing curves, a sweep curve $γ\in Γ$, and a point $P$ in $\tildeγ$, we show that we can continuously shrink $γ$ to $P$ so that throughout the process, the arrangement remains non-piercing (except at a finite set of points in time where $γ$ crosses other curves), and $P$ lies in $\tildeγ$. We show that our arguments can be modified if $P$ lies outside $\tildeγ$, and we want to sweep $γ$ \emph{outwards} so that $P$ lies outside $\tildeγ$, and the arrangement remains non-piercing. As a second contribution, we give an alternate proof of the result of Snoeyink and Hershberger, and give several applications of our results to combinatorial and algorithmic questions including to the \emph{multi-hitting set} problem involving points and non-piercing regions.

cs.CG↗

arXiv2025

Supports for Outerplanar and Bounded Treewidth Graphs

We study the existence and construction of sparse supports for hypergraphs derived from subgraphs of a graph $G$. For a hypergraph $(X,\mathcal{H})$, a support $Q$ is a graph on $X$ s.t. $Q[H]$, the graph induced on vertices in $H$ is connected for every $H\in\mathcal{H}$. We consider \emph{primal}, \emph{dual}, and \emph{intersection} hypergraphs defined by subgraphs of a graph $G$ that are \emph{non-piercing}, (i.e., each subgraph is connected, their pairwise differences remain connected). If $G$ is outerplanar, we show that the primal, dual and intersection hypergraphs admit supports that are outerplanar. For a bounded treewidth graph $G$, we show that if the subgraphs are non-piercing, then there exist supports for the primal and dual hypergraphs of treewidth $O(2^{tw(G)})$ and $O(2^{4tw(G)})$ respectively, and a support of treewidth $2^{O(2^{tw(G)})}$ for the intersection hypergraph. We also show that for the primal and dual hypergraphs, the exponential blow-up of treewidth is sometimes essential. All our results are algorithmic and yield polynomial-time algorithms (when the treewidth is bounded). The existence and construction of sparse supports is a crucial step in the design and analysis of PTASs and/or sub-exponential time algorithms for several packing and covering problems.

math.CO↗

arXiv2025

On Supports for graphs of bounded genus

Let $(X,\mathcal{E})$ 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. We consider the problem of constructing a support for hypergraphs defined by connected subgraphs of a host graph. For a graph $G=(V,E)$, let $\mathcal{H}$ be a set of connected subgraphs of $G$. Let the vertices of $G$ be partitioned into two sets the \emph{terminals} $\mathbf{b}(V)$ and the \emph{non-terminals} $\mathbf{r}(V)$. We define a hypergraph on $\mathbf{b}(V)$, where each $H\in\mathcal{H}$ defines a hyperedge consisting of the vertices of $\mathbf{b}(V)$ in $H$. We also consider the problem of constructing a support for the \emph{dual hypergraph} - a hypergraph on $\mathcal{H}$ where each $v\in \mathbf{b}(V)$ defines a hyperedge consisting of the subgraphs in $\mathcal{H}$ containing $v$. In fact, we construct supports for a common generalization of the primal and dual settings called the \emph{intersection hypergraph}. As our main result, we show that if the host graph $G$ has bounded genus and the subgraphs in $\mathcal{H}$ satisfy a condition of being \emph{cross-free}, then there exists a support that also has bounded genus. Our results are a generalization of the results of Raman and Ray (Rajiv Raman, Saurabh Ray: Constructing Planar Support for Non-Piercing Regions. Discret. Comput. Geom. 64(3): 1098-1122 (2020)). Our techniques imply a unified analysis for packing and covering problems for hypergraphs defined on surfaces of bounded genus. We also describe applications of our results for hypergraph colorings.

math.CO↗

arXiv2025

Adaptive Class Learning to Screen Diabetic Disorders in Fundus Images of Eye

The prevalence of ocular illnesses is growing globally, presenting a substantial public health challenge. Early detection and timely intervention are crucial for averting visual impairment and enhancing patient prognosis. This research introduces a new framework called Class Extension with Limited Data (CELD) to train a classifier to categorize retinal fundus images. The classifier is initially trained to identify relevant features concerning Healthy and Diabetic Retinopathy (DR) classes and later fine-tuned to adapt to the task of classifying the input images into three classes: Healthy, DR, and Glaucoma. This strategy allows the model to gradually enhance its classification capabilities, which is beneficial in situations where there are only a limited number of labeled datasets available. Perturbation methods are also used to identify the input image characteristics responsible for influencing the models decision-making process. We achieve an overall accuracy of 91% on publicly available datasets.

cs.CV↗

arXiv2024

A fast algorithm for computing a planar support for non-piercing rectangles

For a hypergraph $\mathcal{H}=(X,\mathcal{E})$ a \emph{support} is a graph $G$ on $X$ such that for each $E\in\mathcal{E}$, the induced subgraph of $G$ on the elements in $E$ is connected. If $G$ is planar, we call it a planar support. A set of axis parallel rectangles $\mathcal{R}$ forms a non-piercing family if for any $R_1, R_2 \in \mathcal{R}$, $R_1 \setminus R_2$ is connected. Given a set $P$ of $n$ points in $\mathbb{R}^2$ and a set $\mathcal{R}$ of $m$ \emph{non-piercing} axis-aligned rectangles, we give an algorithm for computing a planar support for the hypergraph $(P,\mathcal{R})$ in $O(n\log^2 n + (n+m)\log m)$ time, where each $R\in\mathcal{R}$ defines a hyperedge consisting of all points of $P$ contained in~$R$. We use this result to show that if for a family of axis-parallel rectangles, any point in the plane is contained in at most $k$ pairwise \emph{crossing} rectangles (a pair of intersecting rectangles such that neither contains a corner of the other is called a crossing pair of rectangles), then we can obtain a support as the union of $k$ planar graphs.

cs.CG↗

arXiv2024

The Persistent Robot Charging Problem for Long-Duration Autonomy

This paper introduces a novel formulation aimed at determining the optimal schedule for recharging a fleet of $n$ heterogeneous robots, with the primary objective of minimizing resource utilization. This study provides a foundational framework applicable to Multi-Robot Mission Planning, particularly in scenarios demanding Long-Duration Autonomy (LDA) or other contexts that necessitate periodic recharging of multiple robots. A novel Integer Linear Programming (ILP) model is proposed to calculate the optimal initial conditions (partial charge) for individual robots, leading to the minimal utilization of charging stations. This formulation was further generalized to maximize the servicing time for robots given adequate charging stations. The efficacy of the proposed formulation is evaluated through a comparative analysis, measuring its performance against the thrift price scheduling algorithm documented in the existing literature. The findings not only validate the effectiveness of the proposed approach but also underscore its potential as a valuable tool in optimizing resource allocation for a range of robotic and engineering applications.

cs.RO↗

arXiv2023

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.

math.CO↗

arXiv2021

Quantitative and Qualitative Evaluation of Explainable Deep Learning Methods for Ophthalmic Diagnosis

Background: The lack of explanations for the decisions made by algorithms such as deep learning has hampered their acceptance by the clinical community despite highly accurate results on multiple problems. Recently, attribution methods have emerged for explaining deep learning models, and they have been tested on medical imaging problems. The performance of attribution methods is compared on standard machine learning datasets and not on medical images. In this study, we perform a comparative analysis to determine the most suitable explainability method for retinal OCT diagnosis. Methods: A commonly used deep learning model known as Inception v3 was trained to diagnose 3 retinal diseases - choroidal neovascularization (CNV), diabetic macular edema (DME), and drusen. The explanations from 13 different attribution methods were rated by a panel of 14 clinicians for clinical significance. Feedback was obtained from the clinicians regarding the current and future scope of such methods. Results: An attribution method based on a Taylor series expansion, called Deep Taylor was rated the highest by clinicians with a median rating of 3.85/5. It was followed by two other attribution methods, Guided backpropagation and SHAP (SHapley Additive exPlanations). Conclusion: Explanations of deep learning models can make them more transparent for clinical diagnosis. This study compared different explanations methods in the context of retinal OCT diagnosis and found that the best performing method may not be the one considered best for other deep learning tasks. Overall, there was a high degree of acceptance from the clinicians surveyed in the study. Keywords: explainable AI, deep learning, machine learning, image processing, Optical coherence tomography, retina, Diabetic macular edema, Choroidal Neovascularization, Drusen

eess.IV↗

arXiv2020

Underspecification Presents Challenges for Credibility in Modern Machine Learning

ML models often exhibit unexpectedly poor behavior when they are deployed in real-world domains. We identify underspecification as a key reason for these failures. An ML pipeline is underspecified when it can return many predictors with equivalently strong held-out performance in the training domain. Underspecification is common in modern ML pipelines, such as those based on deep learning. Predictors returned by underspecified pipelines are often treated as equivalent based on their training domain performance, but we show here that such predictors can behave very differently in deployment domains. This ambiguity can lead to instability and poor model behavior in practice, and is a distinct failure mode from previously identified issues arising from structural mismatch between training and deployment domains. We show that this problem appears in a wide variety of practical ML pipelines, using examples from computer vision, medical imaging, natural language processing, clinical risk prediction based on electronic health records, and medical genomics. Our results show the need to explicitly account for underspecification in modeling pipelines that are intended for real-world deployment in any domain.

cs.LG↗

arXiv2018

Deep Learning vs. Human Graders for Classifying Severity Levels of Diabetic Retinopathy in a Real-World Nationwide Screening Program

Deep learning algorithms have been used to detect diabetic retinopathy (DR) with specialist-level accuracy. This study aims to validate one such algorithm on a large-scale clinical population, and compare the algorithm performance with that of human graders. 25,326 gradable retinal images of patients with diabetes from the community-based, nation-wide screening program of DR in Thailand were analyzed for DR severity and referable diabetic macular edema (DME). Grades adjudicated by a panel of international retinal specialists served as the reference standard. Across different severity levels of DR for determining referable disease, deep learning significantly reduced the false negative rate (by 23%) at the cost of slightly higher false positive rates (2%). Deep learning algorithms may serve as a valuable tool for DR screening.

cs.CV↗

arXiv2014

QPTAS for Geometric Set-Cover Problems via Optimal Separators

Weighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal-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 $(1+ε)$-approximability status for most geometric set-cover problems, except for four 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 \emph{quasi-sampling} technique, which together with improvements by Chan \etal~(SODA 2012), yielded a $O(1)$-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in $\Re^3$, 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-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 $\textbf{NP} \nsubseteq \textbf{DTIME}(2^{polylog(n)})$. Together with the recent work of Chan-Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems.

cs.CG↗

arXiv2009

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

In the \emph{tollbooth problem}, we are given a tree $\bT=(V,E)$ with $n$ edges, and a set of $m$ customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of $\bT$ such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when $\bT$ is restricted to be a line. For the tollbooth problem, we present a randomized $O(\log n)$-approximation, improving on the current best $O(\log m)$-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of $\bT$. In this case, we present an algorithm that returns a $(1-ε)$-approximation, for any $ε> 0$, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the \emph{coupon model}, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard.

cs.DS↗