arXiv subjects

Sujoy Bhore

Publications and source records attributed to Sujoy Bhore.

59 records · Page 4Linked to original sources

arXiv2020

Geometric Systems of Unbiased Representatives

Let $P$ be a set of points in $\mathbb{R}^d$, $B$ a bicoloring of $P$ and $\Oo$ a family of geometric objects (that is, intervals, boxes, balls, etc). An object from $\Oo$ is called balanced with respect to $B$ if it contains the same number of points from each color of $B$. For a collection $\B$ of bicolorings of $P$, a geometric system of unbiased representatives (G-SUR) is a subset $\Oo'\subseteq\Oo$ such that for any bicoloring $B$ of $\B$ there is an object in $\Oo'$ that is balanced with respect to $B$. We study the problem of finding G-SURs. We obtain general bounds on the size of G-SURs consisting of intervals, size-restricted intervals, axis-parallel boxes and Euclidean balls. We show that the G-SUR problem is NP-hard even in the simple case of points on a line and interval ranges. Furthermore, we study a related problem on determining the size of the largest and smallest balanced intervals for points on the real line with a random distribution and coloring. Our results are a natural extension to a geometric context of the work initiated by Balachandran et al. on arbitrary systems of unbiased representatives.

cs.CG↗

arXiv2019

Algorithm and Hardness results on Liar's Dominating Set and $k$-tuple Dominating Set

Given a graph $G=(V,E)$, the dominating set problem asks for a minimum subset of vertices $D\subseteq V$ such that every vertex $u\in V\setminus D$ is adjacent to at least one vertex $v\in D$. That is, the set $D$ satisfies the condition that $|N[v]\cap D|\geq 1$ for each $v\in V$, where $N[v]$ is the closed neighborhood of $v$. In this paper, we study two variants of the classical dominating set problem: $\boldmath{k}$-tuple dominating set ($k$-DS) problem and Liar's dominating set (LDS) problem, and obtain several algorithmic and hardness results. On the algorithmic side, we present a constant factor ($\frac{11}{2}$)-approximation algorithm for the Liar's dominating set problem on unit disk graphs. Then, we obtain a PTAS for the $\boldmath{k}$-tuple dominating set problem on unit disk graphs. On the hardness side, we show a $Ω(n^2)$ bits lower bound for the space complexity of any (randomized) streaming algorithm for Liar's dominating set problem as well as for the $\boldmath{k}$-tuple dominating set problem. Furthermore, we prove that the Liar's dominating set problem on bipartite graphs is W[2]-hard.

cs.CG↗

arXiv2019

Balanced Connected Subgraph Problem in Geometric Intersection Graphs

We study the Balanced Connected Subgraph(shortly, BCS) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. Given a vertex-colored graph $G=(V,E)$, where each vertex in $V$ is colored with either ``red'' or ``blue'', the BCS problem seeks a maximum cardinality induced connected subgraph $H$ of $G$ such that $H$ is color-balanced, i.e., $H$ contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithm for the Steiner Tree problem on both the interval graphs and circular arc graphs, that is used as a subroutine for solving BCS problem on same graph classes. Finally, we present a FPT algorithm for the BCS problem on general graphs.

cs.DM↗

arXiv2019

Parameterized Algorithms for Book Embedding Problems

A k-page book embedding of a graph G draws the vertices of G on a line and the edges on k half-planes (called pages) bounded by this line, such that no two edges on the same page cross. We study the problem of determining whether G admits a k-page book embedding both when the linear order of the vertices is fixed, called Fixed-Order Book Thickness, or not fixed, called Book Thickness. Both problems are known to be NP-complete in general. We show that Fixed-Order Book Thickness and Book Thickness are fixed-parameter tractable parameterized by the vertex cover number of the graph and that Fixed-Order Book Thickness is fixed-parameter tractable parameterized by the pathwidth of the vertex order.

cs.DS↗

arXiv2018

The Balanced Connected Subgraph Problem

The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Balanced Connected Subgraph (shortly, BCS) problem. The input is a graph $G=(V,E)$, with each vertex in the set $V$ having an assigned color, "red" or "blue". We seek a maximum-cardinality subset $V'\subseteq V$ of vertices that is color-balanced (having exactly $|V'|/2$ red nodes and $|V'|/2$ blue nodes), such that the subgraph induced by the vertex set $V'$ in $G$ is connected. We show that the BCS problem is NP-hard, even for bipartite graphs $G$ (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph $G$, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue $2$-coloring, and graphs with diameter $2$. For each of these classes either we prove NP-hardness or design a polynomial time algorithm.

cs.DM↗