Searcharxiv⌕ Search

arXiv subjects

Subir Kumar Ghosh

Publications and source records attributed to Subir Kumar Ghosh.

11 recordsLinked to original sources
arXiv2020

On conflict-free chromatic guarding of simple polygons

We study the problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour. The goal is to minimise the number of colours used. This is also known as the conflict-free chromatic guarding problem with vertex guards, and is motivated, e.g., by the problem of radio frequency assignment to sensors placed at the polygon vertices. We first study the scenario in which viewers can be all points of the polygon (such as a mobile robot which moves in the interior of the polygon). We efficiently solve the related problem of minimising the number of guards and approximate (up to only an additive error) the number of colours required in the special case of polygons called funnels. Then we give an upper bound of O(log^2 n) colours on n-vertex weak visibility polygons, by decomposing the problem into sub-funnels. This bound generalises to all simple polygons. We briefly look also at the second scenario, in which the viewers are only the vertices of the polygon. We show a lower bound of 3 colours in the general case of simple polygons and conjecture that this is tight. We also prove that already deciding whether 1 or 2 colours are enough is NP-complete.

cs.CG↗
arXiv2019

Hardness and approximation for the geodetic set problem in some graph classes

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{MGS} problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless $P=NP$, there is no polynomial time algorithm to solve the \textsc{MGS} problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter $2$. On the positive side, we give an $O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS} problem on general graphs of order $n$. We also give a $3$-approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs.

cs.DM↗
arXiv2018

Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards

The art gallery problem enquires about the least number of guards sufficient to ensure that an art gallery, represented by a simple polygon $P$, is fully guarded. Most standard versions of this problem are known to be NP-hard. In 1987, Ghosh provided a deterministic $\mathcal{O}(\log n)$-approximation algorithm for the case of vertex guards and edge guards in simple polygons. In the same paper, Ghosh also conjectured the existence of constant ratio approximation algorithms for these problems. We present here three polynomial-time algorithms with a constant approximation ratio for guarding an $n$-sided simple polygon $P$ using vertex guards. (i) The first algorithm, that has an approximation ratio of 18, guards all vertices of $P$ in $\mathcal{O}(n^4)$ time. (ii) The second algorithm, that has the same approximation ratio of 18, guards the entire boundary of $P$ in $\mathcal{O}(n^5)$ time. (iii) The third algorithm, that has an approximation ratio of 27, guards all interior and boundary points of $P$ in $\mathcal{O}(n^5)$ time. Further, these algorithms can be modified to obtain similar approximation ratios while using edge guards. The significance of our results lies in the fact that these results settle the conjecture by Ghosh regarding the existence of constant-factor approximation algorithms for this problem, which has been open since 1987 despite several attempts by researchers. Our approximation algorithms exploit several deep visibility structures of simple polygons which are interesting in their own right.

cs.CG↗
arXiv2016

Identifying collusion groups using spectral clustering

In an illiquid stock, traders can collude and place orders on a predetermined price and quantity at a fixed schedule. This is usually done to manipulate the price of the stock or to create artificial liquidity in the stock, which may mislead genuine investors. Here, the problem is to identify such group of colluding traders. We modeled the problem instance as a graph, where each trader corresponds to a vertex of the graph and trade corresponds to edges of the graph. Further, we assign weights on edges depending on total volume, total number of trades, maximum change in the price and commonality between two vertices. Spectral clustering algorithms are used on the constructed graph to identify colluding group(s). We have compared our results with simulated data to show the effectiveness of spectral clustering to detecting colluding groups. Moreover, we also have used parameters of real data to test the effectiveness of our algorithm.

q-fin.TR↗
arXiv2016

Approximability of Guarding Weak Visibility Polygons

The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon $P$, is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard $P$ were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for vertex guards and edge guards that achieved a ratio of $\mathcal{O}(\log n)$, which was improved upto $\mathcal{O}(\log\log OPT)$ by King and Kirkpatrick in 2011. It has been conjectured that constant-factor approximation algorithms exist for these problems. We settle the conjecture for the special class of polygons that are weakly visible from an edge and contain no holes by presenting a 6-approximation algorithm for finding the minimum number of vertex guards that runs in $\mathcal{O}(n^2)$ time. On the other hand, for weak visibility polygons with holes, we present a reduction from the Set Cover problem to show that there cannot exist a polynomial time algorithm for the vertex guard problem with an approximation ratio better than $((1 - ε)/12)\ln n$ for any $ε>0$, unless NP=P. We also show that, for the special class of polygons without holes that are orthogonal as well as weakly visible from an edge, the approximation ratio can be improved to 3. Finally, we consider the Point Guard problem and show that it is NP-hard in the case of polygons weakly visible from an edge.

cs.CG↗
arXiv2016

A National Effort for Motivating Indian Students and Teachers towards Algorithmic Research

During 2008-2015, twenty-two introductory workshops on graph and geometric algorithms were organized for teachers and students (undergraduate, post-graduate and doctoral) of engineering colleges and universities at different states and union territories of India. The lectures were meant to provide exposure to the field of graph and geometric algorithms and to motivate the participants towards research. Fifty-eight professors from TIFR, IITs, IISc, IMSc, CMI, ISI Kolkata, and other institutes and universities delivered invited lectures on different topics in the design and analysis of algorithms, discrete applied mathematics, computer graphics, computer vision, and robotics. The first four workshops were funded by TIFR, BRNS and IIT Kharagpur, and the remaining workshops were funded by the NBHM. In this paper, we present the salient features of these workshops, and state our observations on the national impact of these workshops.

cs.CY↗
arXiv2014

An Algorithm for Computing Constrained Reflection Paths in Simple Polygon

Let $s$ be a source point and $t$ be a destination point inside an $n$-vertex simple polygon $P$. Euclidean shortest paths and minimum-link paths between $s$ and $t$ inside $P$ have been well studied. Both these kinds of paths are simple and piecewise-convex. However, computing optimal paths in the context of diffuse or specular reflections does not seem to be an easy task. A path from a light source $s$ to $t$ inside $P$ is called a diffuse reflection path if the turning points of the path lie in the interiors of the boundary edges of $P$. A diffuse reflection path is said to be optimal if it has the minimum number of turning points amongst all diffuse reflection paths between $s$ and $t$. The minimum diffuse reflection path may not be simple. The problem of computing the minimum diffuse reflection path in low degree polynomial time has remained open. In our quest for understanding the geometric structure of the minimum diffuse reflection paths vis-a-vis shortest paths and minimum link paths, we define a new kind of diffuse reflection path called a constrained diffuse reflection path where (i) the path is simple, (ii) it intersects only the eaves of the Euclidean shortest path between $s$ and $t$, and (iii) it intersects each eave exactly once. For computing a minimum constrained diffuse reflection path from $s$ to $t$, we present an $O(n(n+β))$ time algorithm, where $β=Θ(n^2)$ in the worst case. Here, $β$ depends on the shape of the polygon. We also establish some properties relating minimum constrained diffuse reflection paths and minimum diffuse reflection paths. Constrained diffuse reflection paths introduced in this paper provide new geometric insights into the hitherto unknown structures and shapes of optimal reflection paths.

cs.CG↗
arXiv2014

Some Results On Point Visibility Graphs

In this paper, we present three necessary conditions for recognizing point visibility graphs. We show that this recognition problem lies in PSPACE. We state new properties of point visibility graphs along with some known properties that are important in understanding point visibility graphs. For planar point visibility graphs, we present a complete characterization which leads to a linear time recognition and reconstruction algorithm.

cs.CG↗
arXiv2013

Four-connected triangulations of planar point sets

In this paper, we consider the problem of determining in polynomial time whether a given planar point set $P$ of $n$ points admits 4-connected triangulation. We propose a necessary and sufficient condition for recognizing $P$, and present an $O(n^3)$ algorithm of constructing a 4-connected triangulation of $P$. Thus, our algorithm solves a longstanding open problem in computational geometry and geometric graph theory. We also provide a simple method for constructing a noncomplex triangulation of $P$ which requires $O(n^2)$ steps. This method provides a new insight to the structure of 4-connected triangulation of point sets.

cs.CG↗
arXiv2011

On joint triangulations of two sets of points in the plane

In this paper, we establish two necessary conditions for a joint triangulation of two sets of $n$ points in the plane and conjecture that they are sufficient. We show that these necessary conditions can be tested in $O(n^3)$ time. For the problem of a joint triangulation of two simple polygons of $n$ vertices, we propose an $O(n^3)$ time algorithm for constructing a joint triangulation using dynamic programming.

cs.DM↗