arXiv subjects

Ratnadip Mandal

Publications and source records attributed to Ratnadip Mandal.

3 recordsLinked to original sources

arXiv2025

Online Geometric Hitting Set and Set Cover Beyond Unit Balls in $\mathbb{R}^2$

We investigate the geometric hitting set problem in the online setup for the range space $Σ=({\cal P},{\cal S})$, where the set $¶\subset\mathbb{R}^2$ 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 $Σ$ 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 $Θ(\log n)$ 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 $Σ$ 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 $Θ(\log(n))$ 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.

cs.CG↗

arXiv2024

New Lower Bound and Algorithms for Online Geometric Hitting Set Problem

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 $Ω(d\log M)$, whether it is deterministic or random. This improves the recently known deterministic lower bound of $Ω(\log M)$ 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 ${\lfloor\frac{2}α+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ competitive algorithm to hit similarly sized {$α$-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).

cs.CG↗

arXiv2024

Online Class Cover Problem

In this paper, we study the online class cover problem where a (finite or infinite) family $\cal F$ of geometric objects and a set ${\cal P}_r$ of red points in $\mathbb{R}^d$ are given a prior, and blue points from $\mathbb{R}^d$ arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from $\cal F$ that do not cover any points of ${\cal P}_r$. The objective of the problem is to place a minimum number of objects. When $\cal F$ consists of axis-parallel unit squares in $\mathbb{R}^2$, we prove that the competitive ratio of any deterministic online algorithm is $Ω(\log |{\cal P}_r|)$, and also propose an $O(\log |{\cal P}_r|)$-competitive deterministic algorithm for the problem.

cs.CG↗