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Sandip Das

Publications and source records attributed to Sandip Das.

At least 55 records · Page 3Linked to original sources
arXiv2020

On the signed chromatic number of some classes of graphs

A signed graph $(G, σ)$ is a graph $G$ along with a function $σ: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph $(G, σ)$ is the minimum number of vertices $|V(H)|$ of a signed graph $(H, π)$ to which $(G, σ)$ admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention in the last decades, especially due to their strong connections to the theories of graph coloring and graph minors. These homomorphisms have been particularly studied through the scope of the signed chromatic number. In this work, we provide new results and bounds on the signed chromatic number of several families of signed graphs (planar graphs, triangle-free planar graphs, $K_n$-minor-free graphs, and bounded-degree graphs).

cs.DM↗
arXiv2020

Thermal flux in unsteady Rayleigh-Bénard magnetoconvection

We present results of numerical investigation on thermal flux in Rayleigh-Bénard magnetoconvection in the presence of a uniform vertical magnetic field. We have studied thermal flux in different viscous fluids with a range of Prandtl number ($0.1 \le \mathrm{Pr} < 6.5$) and a range of Chandrasekhar number ($50 \le \mathrm{Q} \le 2.5 \times 10^4$). The power spectral density of the Nusselt number varies with frequency $f$ approximately as $f^{-2}$. The probability distribution function of the fluctuating part of the Nusselt number is nearly normal distribution with slight asymmetric tails. For a fixed value the Rayleigh number $\mathrm{Ra}$, the time averaged Nusselt number $\langle \mathrm{Nu} (\mathrm{Q})\rangle$ decreases logarithmically with Chandrasekhar number for $\mathrm{Q} > \mathrm{Q}_c$, which depends on $\mathrm{Ra}$ and $\mathrm{Pr}$. The reduced Nusselt number $\mathrm{Nu_r}$ $=$ $\langle \mathrm{Nu}(\mathrm{Q})\rangle/{\langle \mathrm{Nu}(0)\rangle}$ rises sharply, reaches a maximum slightly above unity and then start decreasing very slowly to unity as the value of a dimensionless parameter $\sqrt{\mathrm{Ra/(Q~Pr)}}$ is raised. The probability distribution function of the local thermal flux in the vertical direction is found to be asymmetric and non-Gaussian with a cusp at its maximum.

physics.flu-dyn↗
arXiv2019

Computation of Spatial Skyline Points

We discuss a method of finding skyline or non-dominated sites in a set $P$ of $n$ point sites with respect to a set $S$ of $m$ points. A site $p \in P$ is non-dominated if and only if for each $q \in P \setminus \{p\}$, there exists at least one point $s \in S$ that is closer to $p$ than to $q$. We reduce this problem of determining non-dominated sites to the problem of finding sites that have non-empty cells in an additively weighted Voronoi diagram under a convex distance function. The weights of said Voronoi diagram are derived from the coordinates of the sites of $P$, while the convex distance function is derived from $S$. In the two-dimensional plane, this reduction gives an $O((n + m) \log (n + m))$-time algorithm to find the non-dominated points.

cs.CG↗
arXiv2019

Pushable chromatic number of graphs with degree constraints

Pushable homomorphisms and the pushable chromatic number $χ_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $χ_p(\overrightarrow{G}) \leq χ_o(\overrightarrow{G}) \leq 2 χ_p(\overrightarrow{G})$, where $χ_o(\overrightarrow{G})$ denotes the oriented chromatic number of $\overrightarrow{G}$. This stands as first general bounds on $χ_p$. This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all $Δ\geq 29$, we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree $Δ$ lies between $2^{\fracΔ{2}-1}$ and $(Δ-3) \cdot (Δ-1) \cdot 2^{Δ-1} + 2$ which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when $Δ\leq 3$, we then prove that the maximum value of the pushable chromatic number is~$6$ or~$7$. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~$3$ lies between~$5$ and~$6$. The former upper bound of~$7$ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~$6$.

cs.DM↗
arXiv2019

A Variable Rate Fronthaul Scheme for Cloud Radio Access Networks (C-RAN)

Cloud Radio Access Network (C-RAN) is considered as one of the most promising candidate for Fifth Generation (5G) wireless communication system. However, C-RAN imposes stringent requirements on fronthaul in terms of capacity which in turn becomes a bottleneck for deployment of cloud-RAN in dense networks. Compressed CPRI and functional split are two primary solutions to overcome this. However, this makes Remote Radio Unit (RRU) more complex and the computational resources in RRU are localized and thus cannot be shared with other RRUs when not in use. In this paper, we introduce the concept of Variable Rate Fronthaul (VRF) for C-RAN. This scheme operates on a CPRI type of interface (e.g., one that transmits I/Q data samples) with the novelty of dynamically changing the cell bandwidth, and consequently the fronthaul data rates, depending on the cell load, with the support of a Software Defined Network (SDN) controller. This allows for a more efficient transport of C-RAN cells data over a shared backhaul. We first propose a mathematical analysis of the VRF performance using a queuing theory approach based on the Markov model. We then provide the results of our simulation framework both for validation and in support of the mathematical analysis. Our results show that the proposed VRF scheme provides significantly lower blocking probability over a shared backhaul than standard CPRI.

eess.SP↗
arXiv2018

Computing the Minkowski Sum of Convex Polytopes in $\Re^d$

We propose a method to efficiently compute the Minkowski sum, denoted by binary operator $\oplus$ in the paper, of convex polytopes in $\Re^d$ using their face lattice structures as input. In plane, the Minkowski sum of convex polygons can be computed in linear time of the total number of vertices of the polygons. In $\Re^d$, we first show how to compute the Minkowski sum, $P \oplus Q$, of two convex polytopes $P$ and $Q$ of input size $n$ and $m$ respectively in time $O(nm)$. Then we generalize the method to compute the Minkowski sum of $n$ convex polytopes, $P_1 \oplus P_2 \oplus \cdots \oplus P_n$, in $\Re^d$ in time $O(\prod_{i}^{n}N_i)$, where $P_1$, $P_2$, $\dots$, $P_n$ are $n$ input convex polytopes and for each $i$, $N_i$ is size of the face lattice structure of $P_i$. Our algorithm for Minkowski sum of two convex polytopes is optimal in the worst case since the output face lattice structure of $P\oplus Q$ for convex polytopes in $\Re^d$ can be $O(nm)$ in worst case.

cs.CG↗
arXiv2018

On relative clique number of colored mixed graphs

An $(m, n)$-colored mixed graph is a graph having arcs of $m$ different colors and edges of $n$ different colors. A graph homomorphism of an $(m, n$)-colored mixed graph $G$ to an $(m, n)$-colored mixed graph $H$ is a vertex mapping such that if $uv$ is an arc (edge) of color $c$ in $G$, then $f(u)f(v)$ is also an arc (edge) of color $c$. The ($m, n)$-colored mixed chromatic number of an $(m, n)$-colored mixed graph $G$, introduced by Nešetřil and Raspaud [J. Combin. Theory Ser. B 2000] is the order (number of vertices) of the smallest homomorphic image of $G$. Later Bensmail, Duffy and Sen [Graphs Combin. 2017] introduced another parameter related to the $(m, n)$-colored mixed chromatic number, namely, the $(m, n)$-relative clique number as the maximum cardinality of a vertex subset which, pairwise, must have distinct images with respect to any colored homomorphism. In this article, we study the $(m, n$)-relative clique number for the family of subcubic graphs, graphs with maximum degree $Δ$, planar graphs and triangle-free planar graphs and provide new improved bounds in each of the cases. In particular, for subcubic graphs we provide exact value of the parameter.

cs.DM↗
arXiv2018

Approximating Minimum Dominating Set on String Graphs

In this paper, we give approximation algorithms for the \textsc{Minimum Dominating Set (MDS)} problem on \emph{string} graphs and its subclasses. A \emph{path} is a simple curve made up of alternating horizontal and vertical line segments. A \emph{$k$-bend path} is a path made up of at most $k + 1$ line segments. An \textsc{L}-path is a $1$-bend path having the shape `\textsc{L}'. A \emph{vertically-stabbed-\textsc{L} graph} is an intersection graph of \textsc{L}-paths intersecting a common vertical line. We give a polynomial time $8$-approximation algorithm for \textsc{MDS} problem on vertically-stabbed-\textsc{L} graphs whose APX-hardness was shown by Bandyapadhyay et al. (\textsc{MFCS}, 2018). To prove the above result, we needed to study the \emph{Stabbing segments with rays} (\textsc{SSR}) problem introduced by Katz et al. (\textsc{Comput. Geom. 2005}). In the \textsc{SSR} problem, the input is a set of (disjoint) leftward-directed rays, and a set of (disjoint) vertical segments. The objective is to select a minimum number of rays that intersect all vertical segments. We give a $O((n+m)\log (n+m))$-time $2$-approximation algorithm for the \textsc{SSR} problem where $n$ and $m$ are the number of rays and segments in the input. A \emph{unit $k$-bend path} is a $k$-bend path whose segments are of unit length. A graph is a \emph{unit $B_k$-VPG graph} if it is an intersection graph of unit $k$-bend paths. Any string graph is a unit-$B_k$-VPG graph for some finite $k$. Using our result on \textsc{SSR}-problem, we give a polynomial time $O(k^4)$-approximation algorithm for \textsc{MDS} problem on unit $B_k$-VPG graphs for $k\geq 0$.

cs.DM↗
arXiv2018

On bounds on bend number of split and cocomparability graphs

A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A $k$-bend path is a path made up of at most $k + 1$ line segments. A $B_k$-VPG representation of a graph is a collection of $k$-bend paths such that each path in the collection represents a vertex of the graph and two such paths intersect if and only if the vertices they represent are adjacent in the graph. The graphs that have a $B_k$-VPG representation are called $B_k$-VPG graphs. It is known that the poset dimension $dim(G)$ of a cocomparability graph $G$ is greater than or equal to its bend number $bend(G)$. Cohen et al. ({\textsc{order 2015}}) asked for examples of cocomparability graphs with low bend number and high poset dimension. We answer this question by proving that for each $m, t \in \mathbb{N}$, there exists a cocomparability graph $G_{t,m}$ with $t < bend(G_{t,m}) \leq 4t+29$ and $dim(G_{t,m})-bend(G_{t,m})>m$. Techniques used to prove the above result, allows us to partially address the open question posed by Chaplick et al. ({\textsc{wg 2012}}) who asked whether $B_k$-VPG-chordal $\subsetneq$ $B_{k+1}$-VPG-chordal for all $k \in \mathbb{N}$. We address this by proving that there are infinitely many $m \in \mathbb{N}$ such that $B_m$-VPG-split $\subsetneq$ $B_{m+1}$-VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all $t \in \mathbb{N}$, $B_t$-VPG-$Forb(C_{\geq 5})$ $\subsetneq$ $B_{4t+29}$-VPG-$Forb(C_{\geq 5})$, where $Forb(C_{\geq 5})$ denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all $t \in \mathbb{N}$, $PB_t$-VPG-split $\subsetneq PB_{36t+80}$-VPG-split, where $PB_t$-VPG denotes the class of graphs with proper bend number at most $t$.

cs.DM↗
arXiv2016

Linear-Time Fitting of a $k$-Step Function

Given a set of $n$ weighted points on the $x$-$y$ plane, we want to find a step function consisting of $k$ horizontal steps such that the maximum vertical weighted distance from any point to a step is minimized. We solve this problem in $O(n)$ time when $k$ is a constant. Our approach relies on the prune-and-search technique, and can be adapted to design similar linear time algorithms to solve the line-constrained k-center problem and the size-$k$ histogram construction problem as well.

cs.CG↗
arXiv2016

Geometric k-Center Problems with Centers Constrained to Two Lines

We consider the $k$-center problem in which the centers are constrained to lie on two lines. Given a set of $n$ weighted points in the plane, we want to locate up to $k$ centers on two parallel lines. We present an $O(n\log^2 n)$ time algorithm, which minimizes the weighted distance from any point to a center. We then consider the unweighted case, where the centers are constrained to be on two perpendicular lines. Our algorithms run in $O(n\log^2 n)$ time also in this case.

cs.CG↗
arXiv2016

The $p$-Center Problem in Tree Networks Revisited

We present two improved algorithms for weighted discrete $p$-center problem for tree networks with $n$ vertices. One of our proposed algorithms runs in $O(n \log n + p \log^2 n \log(n/p))$ time. For all values of $p$, our algorithm thus runs as fast as or faster than the most efficient $O(n\log^2 n)$ time algorithm obtained by applying Cole's speed-up technique [cole1987] to the algorithm due to Megiddo and Tamir [megiddo1983], which has remained unchallenged for nearly 30 years. Our other algorithm, which is more practical, runs in $O(n \log n + p^2 \log^2(n/p))$ time, and when $p=O(\sqrt{n})$ it is faster than Megiddo and Tamir's $O(n \log^2n \log\log n)$ time algorithm [megiddo1983].

cs.DS↗
arXiv2016

Chromatic number of signed graphs with bounded maximum degree

A signed graph $ (G, Σ)$ is a graph positive and negative ($Σ$ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, Σ)$ is to switch the signs of the edges incident to $v$. If one can obtain $ (G, Σ')$ by re-signing some vertices of $ (G, Σ)$, then $ (G, Σ) \equiv (G, Σ')$. A signed graphs $ (G, Σ)$ admits an homomorphism to $ (H, Λ)$ if there is a sign preserving vertex mapping from $(G,Σ')$ to $(H, Λ)$ for some $ (G, Σ) \equiv (G, Σ')$. The signed chromatic number $χ_{s}( (G, Σ))$ of the signed graph $(G, Σ)$ is the minimum order (number of vertices) of a signed graph $(H, Λ)$ such that $ (G, Σ)$ admits a homomorphism to $(H, Λ)$. For a family $ \mathcal{F}$ of signed graphs $χ_{s}(\mathcal{F}) = \text{max}_{(G,Σ) \in \mathcal{F}} χ_{s}( (G, Σ))$. We prove $2^{Δ/2-1} \leq χ_s(\mathcal{G}_Δ) \leq (Δ-1)^2. 2^{(Δ-1)} +2$ for all $Δ\geq 3$ where $\mathcal{G}_Δ$ is the family of connected signed graphs with maximum degree $Δ$. \end{abstract}

math.CO↗
arXiv2016

On a special class of boxicity 2 graphs

We define and study a class of graphs, called 2-stab interval graphs (2SIG), with boxicity 2 which properly contains the class of interval graphs. A 2SIG is an axes-parallel rectangle intersection graph where the rectangles have unit height (that is, length of the side parallel to $Y$-axis) and intersects either of the two fixed lines, parallel to the $X$-axis, distance $1+ε$ ($0 < ε< 1$) apart. Intuitively, 2SIG is a graph obtained by putting some edges between two interval graphs in a particular rule. It turns out that for these kind of graphs, the chromatic number of any of its induced subgraphs is bounded by twice of its (induced subgraph) clique number. This shows that the graph, even though not perfect, is not very far from it. Then we prove similar results for some subclasses of 2SIG and provide efficient algorithm for finding their clique number. We provide a matrix characterization for a subclass of 2SIG graph.

cs.DM↗
arXiv2016

On local structures of cubicity 2 graphs

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + ε$ ($0 < ε< 1$) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation.

cs.DM↗
arXiv2015

On chromatic number of colored mixed graphs

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph $H$ is a vertex mapping such that if $uv$ is an arc (edge) of color $c$ in $G$, then $f(u)f(v)$ is an arc (edge) of color $c$ in $H$. The \textit{$(m,n)$-colored mixed chromatic number} $χ_{(m,n)}(G)$ of an $(m,n)$-colored mixed graph $G$ is the order (number of vertices) of the smallest homomorphic image of $G$. This notion was introduced by Nešetřil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147--155). They showed that $χ_{(m,n)}(G) \leq k(2m+n)^{k-1}$ where $G$ is a $k$-acyclic colorable graph. We proved the tightness of this bound. We also showed that the acyclic chromatic number of a graph is bounded by $k^2 + k^{2 + \lceil log_{(2m+n)} log_{(2m+n)} k \rceil}$ if its $(m,n)$-colored mixed chromatic number is at most $k$. Furthermore, using probabilistic method, we showed that for graphs with maximum degree $Δ$ its $(m,n)$-colored mixed chromatic number is at most $2(Δ-1)^{2m+n} (2m+n)^{Δ-1}$. In particular, the last result directly improves the upper bound $2Δ^2 2^Δ$ of oriented chromatic number of graphs with maximum degree $Δ$, obtained by Kostochka, Sopena and Zhu (1997, J. Graph Theory 24, 331--340) to $2(Δ-1)^2 2^{Δ-1}$. We also show that there exists a graph with maximum degree $Δ$ and $(m,n)$-colored mixed chromatic number at least $(2m+n)^{Δ/ 2}$.

cs.DM↗
arXiv2015

Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers $c, s \geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ points in the plane, no three on a line and colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. The case $s=0$, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=\infty$, for $c\geq 3$. In this paper we extend these results when $c \geq 2$ and $s \geq 1$. We prove that the least integer $λ_3(c)$ such that $M_3(c, λ_3(c))< \infty$ satisfies: $$\left\lfloor\frac{c-1}{2}\right\rfloor \leqλ_3(c)\leq c-2,$$ where $c \geq 2$. Moreover, the exact values of $M_3(c, s)$ are determined for small values of $c$ and $s$. We also conjecture that $λ_3(4)=1$, and verify it for sufficiently large Horton sets.

math.CO↗